Short answer
No. Islam forbids both halves of the act. Suicide is forbidden. The Qur'an says "Nor kill (or destroy) yourselves". The Prophet Muhammad (ﷺ) warned that whoever kills himself will be punished in the Fire by the same means. Killing the innocent is forbidden too. The Qur'an forbids taking any life Allah (﷾) has made sacred, except by justice and law. The Prophet (ﷺ) forbade the killing of women and children, even in a lawful war. He said that whoever attacks the Muslim community, killing good and bad alike and breaking pledges of safety, has nothing to do with him. Bombs in markets, hijackings and hostage-taking all fall under the same ban. Nothing in the words or conduct of the Prophet (ﷺ) or his companions allows them. In Islamic law war belongs to the ruler of a state alone, so no individual or group may launch any attack, on soldiers or on civilians.
Full explanation
Suicide is forbidden
The Qur'an tells believers not to kill themselves, and reminds them that Allah (﷾) has been merciful to them. The Prophet (ﷺ) warned that whoever kills himself will go on doing the same to himself in the Fire. This is so whether he used a fall, poison or a blade. A bomber who kills himself commits this sin, whatever else he does.
“O ye who believe! Eat not up your property among yourselves in vanities: But let there be amongst you Traffic and trade by mutual good-will: Nor kill (or destroy) yourselves: for verily Allah hath been to you Most Merciful!”
Abu Hurayra reports that the Prophet (ﷺ) said whoever kills himself, whether by a fall, by poison or with a blade, will suffer the same in the Fire.
Killing the innocent is forbidden
The Qur'an says that killing a person, unless for murder or for spreading corruption in the land, is like killing all mankind. It forbids taking a life that Allah (﷾) has made sacred except by justice and law. It promises Hell and Allah's (﷾) curse to whoever kills a believer on purpose. Believers are very often the victims of these bombs. Even in a lawful war the Prophet (ﷺ) forbade the killing of women and children. He warned that whoever kills a person under treaty with the Muslims will not smell Paradise.
“On that account: We ordained for the Children of Israel that if any one slew a person - unless it be for murder or for spreading mischief in the land - it would be as if he slew the whole people: and if any one saved a life, it would be as if he saved the life of the whole people. Then although there came to them Our messengers with clear signs, yet, even after that, many of them continued to commit excesses in the land.”
“Say: "Come, I will rehearse what Allah hath (really) prohibited you from": Join not anything as equal with Him; be good to your parents; kill not your children on a plea of want;- We provide sustenance for you and for them;- come not nigh to shameful deeds. Whether open or secret; take not life, which Allah hath made sacred, except by way of justice and law: thus doth He command you, that ye may learn wisdom.”
“Nor take life - which Allah has made sacred - except for just cause. And if anyone is slain wrongfully, we have given his heir authority (to demand qisas or to forgive): but let him not exceed bounds in the matter of taking life; for he is helped (by the Law).”
“If a man kills a believer intentionally, his recompense is Hell, to abide therein (For ever): And the wrath and the curse of Allah are upon him, and a dreadful penalty is prepared for him.”
Ibn Umar reports that a woman was found killed during one of the Prophet's (ﷺ) campaigns, and the Prophet forbade the killing of women and children.
Abdullah ibn Amr reports that the Prophet (ﷺ) said that whoever kills a person who has a treaty with the Muslims will not smell the fragrance of Paradise, though its fragrance reaches a distance of forty years.
The Prophet (ﷺ) disowned such men in advance
The Prophet (ﷺ) described the man who attacks the Muslim community. He strikes the righteous and the wicked alike, spares not even the believer, and breaks faith with those given a pledge of safety. The Prophet (ﷺ) said that such a man has nothing to do with him. He also said that a believer keeps room in his religion until he sheds blood unlawfully. He said that bloodshed will be the first matter judged between people on the Day of Resurrection. The man who kills the innocent in the name of Islam is committing one of its gravest sins.
Abu Hurayra reports that the Prophet (ﷺ) said: whoever attacks his community, striking down the good and the wicked alike, sparing not even the believers, and not keeping faith with those given a pledge of security, has nothing to do with me, and I have nothing to do with him.
Ibn Umar reports that the Prophet (ﷺ) said a believer keeps room in his religion as long as he does not shed blood unlawfully.
Abdullah ibn Mas'ud reports that the Prophet (ﷺ) said the first cases to be judged between people on the Day of Resurrection will be cases of bloodshed.
Fighting has limits, and only a lawful authority wages it
Islam permits a lawful state to fight those who fight it, and forbids going past the limits. The Prophet (ﷺ) called the ruler a shield behind whom Muslims fight. A private man cannot decide for himself whom to kill or punish. War and punishment belong to a lawful authority and a qualified judge. No individual or group may launch any attack of its own, on soldiers or on civilians. A bomb in a market breaks every one of these rules.
“Fight in the cause of Allah those who fight you, but do not transgress limits; for Allah loveth not transgressors.”
Abu Huraira reports that the Prophet (ﷺ) said: whoever obeys the ruler obeys me, and whoever disobeys him disobeys me. The ruler (imam) is a shield: fighting is done behind him and protection is sought through him. If he commands piety and rules justly he is rewarded, and if not, the burden is on him.
Ibn Qudamah, a Hanbali jurist, holds that the matter of jihad is entrusted to the ruler (imam) and his judgement, and that the people must obey him in it. He makes one exception: if an enemy attacks suddenly and there is no time to ask the ruler, the people may defend themselves without his permission. He also holds that a Muslim who enters a non-Muslim land under a promise of safety (aman) may not betray its people or take their property, because the safety was given on the understanding that he would not betray them, even if this was not said in words.
The Bible's hero of faith
The Bible does tell the story of a man who killed himself in order to kill a house full of people. Samson prayed for strength and said "Let me die with the Philistines". He pulled down the house on himself and everyone in it, with about three thousand men and women on the roof alone. The New Testament then names Samson among those who through faith subdued kingdoms and wrought righteousness. The Qur'an and the Prophet (ﷺ) forbid both halves of that act: the suicide, and the killing of women and the innocent.
“Now the house was full of men and women; and all the lords of the Philistines were there; and there were upon the roof about three thousand men and women, that beheld while Samson made sport. And Samson called unto the Lord, and said, O Lord God, remember me, I pray thee, and strengthen me, I pray thee, only this once, O God, that I may be at once avenged of the Philistines for my two eyes. And Samson took hold of the two middle pillars upon which the house stood, and on which it was borne up, of the one with his right hand, and of the other with his left. And Samson said, Let me die with the Philistines. And he bowed himself with all his might; and the house fell upon the lords, and upon all the people that were therein. So the dead which he slew at his death were more than they which he slew in his life.”
“And what shall I more say? for the time would fail me to tell of Gedeon, and of Barak, and of Samson, and of Jephthae; of David also, and Samuel, and of the prophets: Who through faith subdued kingdoms, wrought righteousness, obtained promises, stopped the mouths of lions,”
Verse by verse
Qur'an 5:32 — Surat al-Ma'ida (The Table), verse 32
“On that account: We ordained for the Children of Israel that if any one slew a person - unless it be for murder or for spreading mischief in the land - it would be as if he slew the whole people: and if any one saved a life, it would be as if he saved the life of the whole people. Then although there came to them Our messengers with clear signs, yet, even after that, many of them continued to commit excesses in the land.”
Cited for
The verse about killing one person being like killing all mankind was addressed to the Children of Israel, so it does not bind Muslims.
The response
The Qur'an says the same thing to Muslims directly, more than once. It forbids taking a life Allah (﷾) has made sacred except by justice and law. It tells believers not to kill themselves. It promises Hell to whoever kills a believer on purpose.
Where it lands — The ban on killing the innocent is addressed to Muslims in their own words, so the reply gives the bomber no way out.
Sahih Muslim 1901 — Kitab al-Imara (The Book on Government), hadith 1901
Anas ibn Malik reports that at Badr, when the Prophet (ﷺ) urged his companions to rise to a Paradise as wide as the heavens and the earth, Umayr ibn al-Humam threw away the dates he was eating, fought the enemy and was killed.
Cited for
Companions charged into the enemy's ranks knowing they would be killed, as Umayr ibn al-Humam did at Badr. Modern fatwas (rulings) on "martyrdom operations" point to such reports. So, they say, a man may give his life to strike the enemy.
The response
A soldier who risks death fighting enemy soldiers in a lawful battle is not a man who kills himself. He does not target the innocent. Umayr was a soldier in a battle the Prophet (ﷺ) himself led as head of state, and he was killed by the enemy he fought. That gives no one today a licence: war is for a lawful state alone, so no individual or group may launch any attack, on soldiers or on civilians. The bomber also kills himself and aims at the innocent, and both are forbidden in plain words.
Where it lands — Courage in battle is not a licence for suicide or for killing the innocent.
Sahih al-Bukhari 3012 — Kitab al-Jihad wa al-Siyar (Book of Fighting for the Cause of Allah), hadith 3012
al-Sa'b ibn Jaththama reports that the Prophet (ﷺ) was asked about attacking pagan fighters at night when their women and children might be exposed to harm, and replied that they are from them.
Cited for
Critics say Bukhari 3012 shows civilians may be killed along with fighters.
The response
The report is about a night attack on fighters. Women and children could not be told apart there and might be hit by accident. It gives no leave to target them. The deliberate killing of women and children stayed forbidden. The bomber in a market or on a bus chooses the innocent as his target.
Where it lands — An accidental risk in a night battle with fighters is not the deliberate targeting of civilians.
Hebrews 11:32-33 — Epistle to the Hebrews, chapter 11, verses 32-33
“And what shall I more say? for the time would fail me to tell of Gedeon, and of Barak, and of Samson, and of Jephthae; of David also, and Samuel, and of the prophets: Who through faith subdued kingdoms, wrought righteousness, obtained promises, stopped the mouths of lions,”
Cited for
Christians read Hebrews 11 as praising Samson's faith, not approving every act, and his target was the Philistine lords who had enslaved and blinded him.
The response
The house held about three thousand men and women on the roof alone, and Samson asked to die with them. Scripture records his prayer as answered and names him among the heroes of faith, without a word against the act.
Where it lands — The Bible honours the man; the Qur'an forbids both the suicide and the killing of women.
Perspectives
The Islamic perspective
Suicide bombing is forbidden twice over: suicide is a grave sin, and killing the innocent is one of the gravest. The Prophet (ﷺ) disowned whoever strikes the righteous and the wicked alike and breaks pledges of safety.
The Qur'an forbids believers to kill themselves.
The Prophet (ﷺ) warned that whoever kills himself will be punished in the Fire by the same means.
The Qur'an forbids taking a life Allah (﷾) has made sacred except by justice and law. It likens one unjust killing to the killing of all mankind.
The Prophet (ﷺ) forbade killing women and children in war. He warned that whoever kills a person under treaty will not smell Paradise.
The Prophet (ﷺ) described a man who attacks the Muslim community, killing the righteous and the wicked alike and breaking pledges of safety. He said such a man has nothing to do with him.
Unlawful bloodshed is among the gravest sins in Islam. Cases of bloodshed are the first to be judged between people on the Day of Resurrection.
The New Testament names Samson among the heroes of faith. He died bringing a house down on about three thousand men and women.