Short answer
Yes. The Qur'an says hearts find satisfaction in the remembrance of God (﷾). It says a life turned away from Him is a narrowed life. Islam brings peace in three ways. It gives each person a direct bond with God (﷾), with no priest in between. It teaches that every hardship is known to God (﷾) and can become a reward. And it frees a person from chasing money and fame, which never satisfy. The Prophet Muhammad (ﷺ) said true riches are a contented soul. Research on faith and health points the same way: most studies link religious faith with more well-being and less depression. Cat Stevens was one of the best-known singers of the 1970s. Yet he turned to God (﷾), became a Muslim in 1977 and walked away from his pop career until 2006.
Full explanation
Hearts find satisfaction in the remembrance of God (﷾)
The Qur'an names the source of inner peace plainly: the remembrance of God (﷾). It also names the opposite. Whoever turns away from God's (﷾) message has a narrow, cramped life. God (﷾) sends calm into the hearts of believers. And He promises a good life to every man or woman who believes and does good.
“"Those who believe, and whose hearts find satisfaction in the remembrance of Allah: for without doubt in the remembrance of Allah do hearts find satisfaction.”
“"But whosoever turns away from My Message, verily for him is a life narrowed down, and We shall raise him up blind on the Day of Judgment."”
“It is He Who sent down tranquillity into the hearts of the Believers, that they may add faith to their faith;- for to Allah belong the Forces of the heavens and the earth; and Allah is Full of Knowledge and Wisdom;-”
“Whoever works righteousness, man or woman, and has Faith, verily, to him will We give a new Life, a life that is good and pure and We will bestow on such their reward according to the best of their actions.”
Peace in the very name
The word Islam comes from the same Arabic root, s-l-m, as salam, which means peace. Islam means giving oneself to God (﷾). The Qur'an promises that whoever does so, and does good, will have no fear and no grief. There is no priest or go-between: God (﷾) is nearer to each person than his own jugular vein.
“Nay,-whoever submits His whole self to Allah and is a doer of good,- He will get his reward with his Lord; on such shall be no fear, nor shall they grieve.”
“It was We Who created man, and We know what dark suggestions his soul makes to him: for We are nearer to him than (his) jugular vein.”
Hardship without despair
Islam does not promise a life without trouble. It teaches that every loss is known to God (﷾) before it happens, so the believer does not despair over what passes him by. Those who are patient in loss and say "To Allah we belong, and to Him is our return" receive blessings and mercy. With every hardship comes ease. The Prophet (ﷺ) said the believer's affair is all good. He is thankful in good times and patient in bad, and both are good for him.
“No misfortune can happen on earth or in your souls but is recorded in a decree before We bring it into existence: That is truly easy for Allah: In order that ye may not despair over matters that pass you by, nor exult over favours bestowed upon you. For Allah loveth not any vainglorious boaster,-”
“Be sure we shall test you with something of fear and hunger, some loss in goods or lives or the fruits (of your toil), but give glad tidings to those who patiently persevere, Who say, when afflicted with calamity: "To Allah We belong, and to Him is our return":- They are those on whom (Descend) blessings from Allah, and Mercy, and they are the ones that receive guidance.”
“So, verily, with every difficulty, there is relief: Verily, with every difficulty there is relief.”
Suhayb reports that the Prophet (ﷺ) said the affair of the believer is wonderful, for there is good for him in all of it, and this is so for no one but the believer: if something pleasing happens he gives thanks, and that is good for him; if hardship strikes he is patient, and that is good for him.
Contentment, not possessions
The Prophet (ﷺ) taught that being rich is not having many things. It is having a soul that is content. The Qur'an ends its picture of the soul at peace with God's (﷾) call to it: come back to your Lord, well pleased, and pleasing to Him.
Abu Hurayra reports that the Prophet (ﷺ) said riches do not lie in having many possessions; true riches are contentment of the soul.
“(To the righteous soul will be said:) "O (thou) soul, in (complete) rest and satisfaction! "Come back thou to thy Lord,- well pleased (thyself), and well-pleasing unto Him! "Enter thou, then, among My devotees! "Yea, enter thou My Heaven!”
What research on faith and well-being finds
A large review of studies up to 2010 found that most link religious involvement with greater well-being and less depression. It also found that most of the strongest studies on suicide find less suicide among the more religious. These studies are about religious faith in general, not Islam alone. They fit what the Qur'an teaches: a heart turned to God (﷾) is steadier.
A systematic review of peer-reviewed studies up to 2010. Of 326 studies on well-being, 256 (79%) found only positive links with religious involvement. Of 444 studies on depression, 272 (61%) found less depression among the more religious. Of the 49 most rigorous studies on suicide, 39 (80%) found less suicide, fewer attempts or more negative attitudes to suicide among the more religious. The studies cover religion in general, not Islam alone.
A famous example: Cat Stevens
Cat Stevens rose from teen idol in the 1960s to one of the best-known singer-songwriters of the 1970s. At the height of his fame, in the mid-1970s, he nearly drowned off Malibu, California. He called on God (﷾) and promised to serve Him if he was saved. After his brother gave him a copy of the Qur'an, he became a Muslim in 1977. The next year he took the name Yusuf Islam, and he walked away from the fame he had chased. He did not make a full return to music until 2006.
Records that Cat Stevens grew from a teen idol of the 1960s into one of the most influential singer-songwriters of all time, with songs such as "Morning Has Broken", "Peace Train" and "Moonshadow", that in the mid-1970s, nearly drowning off Malibu, he called on God and promised to dedicate his life to God's service if he was saved, and that after receiving a copy of the Qur'an from his brother he embraced Islam in 1977, took the name Yusuf Islam in 1978, and walked away from fame and his career as a music star, and that his full return to music making came in 2006.
Perspectives
Peace needs no religion
Many people without a religion hold that peace of mind comes from a good life here: love, work, friendship and thought. Bertrand Russell wrote that happiness, thought and love keep their value, even though they come to an end.
Happiness, thought and love keep their value though they end at death, so peace needs no God.
Rebuttal
Love and friendship are real goods. But a peace that ends at the grave lives in its shadow. The studies point the other way too. A review of hundreds of studies found that most of them linked religious practice with greater well-being and less depression. The Qur'an names the source: in the remembrance of Allah (﷾) do hearts find satisfaction.
Hindu: grief passes when the self sees the Lord
The Mundaka Upanishad pictures two birds on one tree. One is the self, grieving and confused. The other is the Lord, at peace. When the self sees the Lord and knows his glory, its grief passes away.
When a man sees the other, the Lord, contented, and knows his glory, his grief passes away.
Rebuttal
Islam agrees that peace comes from knowing the Lord. The Qur'an says hearts find satisfaction in the remembrance of Allah (﷾). The picture of the two birds sets the self and the Lord side by side. Islam keeps them apart. The self never becomes the Lord, and there is none like Him. Islam asks only that the self turn to that one Lord alone.
Buddhist: extinguishment is the highest happiness
The Buddha taught that peace comes when craving ends. The Dhammapada calls contentment the greatest wealth, and calls nibbana, the blowing out of craving, the highest happiness.
Contentment is the ultimate wealth, and extinguishment the ultimate happiness.
Rebuttal
Islam honours contentment too: the Prophet (ﷺ) said true riches are contentment of the soul. But Islam does not ask the heart to put out its longing. It gives that longing its true object: in the remembrance of Allah (﷾) do hearts find satisfaction.
Shinto: purity brings a clear heart
In Shinto, washing away uncleanness through the rites of misogi and harae makes a person new before the kami, the spirits. A pure, sincere heart is the ground of a good and peaceful life.
Ablution washes away misfortune and impurity, and leaves a person ready to meet the kami.
Rebuttal
Islam knows the peace of washing well: the Muslim washes before prayer, as the Qur'an commands. But a clean body is only the door. The Qur'an says what lies beyond it: in the remembrance of Allah (﷾) do hearts find satisfaction.
Daoist: stillness and contentment
The Tao Te Ching teaches peace through emptiness and stillness, guarded with tireless care. And it teaches mastery of yourself, rather than mastery of others.
The state of vacancy should be brought to the utmost degree, and stillness guarded with unwearying vigour.
Rebuttal
Stillness quiets the mind, and Islam values it. The Prophet (ﷺ) taught that true riches are a soul that is content. But a still mind with no one to turn to is only quiet. The Qur'an gives the heart Someone to turn to: in the remembrance of Allah (﷾) do hearts find satisfaction.
He who overcomes others is strong; he who overcomes himself is mighty.
Rebuttal
Islam agrees that mastering yourself is the greater strength. It adds a help from beyond yourself. The believer is thankful in good times and patient in hard times. So everything that happens to him is good for him.
Confucian: self-mastery and the rites
Confucius taught that perfect goodness is to master yourself and return to the rites. Peace comes from a well-ordered self, family and society, and from the joy of learning.
To subdue one's self and return to propriety is perfect virtue.
Rebuttal
Islam agrees that self-mastery and good order bring peace. Prayer itself is a discipline, five times a day. But order among people does not quiet the heart's longing for its Maker. When Confucius was very ill, a disciple asked leave to pray for him. He answered that his own praying had gone on for a long time. The Qur'an says: in the remembrance of Allah (﷾) do hearts find satisfaction.
Chinese tradition: good fortune and a family at peace
For many Chinese families peace means a harmonious home, health and good fortune. People visit their ancestors' graves and pray at temples for wealth and luck.
Peace comes from a harmonious family and good fortune, so people pray at temples for wealth and luck.
Rebuttal
Wealth and luck come and go, and a peace that rests on them goes with them. The Prophet (ﷺ) taught that true riches are a soul that is content, not many possessions. The Qur'an says God (﷾) knows every loss and gain before it happens. That is so people do not despair over what passes them by, nor boast over what comes to them.
The Islamic perspective
Real happiness and peace come from giving yourself to the Creator and remembering Him, as the Qur'an and the Prophet (ﷺ) teach. Money and success cannot give it. A heart at rest with God (﷾) can bear any loss.
The Qur'an says hearts find satisfaction in the remembrance of God (﷾).
The Qur'an says whoever turns away from God's (﷾) message has a narrowed life.
God (﷾) promises a good and pure life to every believer, man or woman, who does good.
Islam gives a direct relationship with God (﷾), with no go-between. It promises no fear or grief to the one who submits to Him and does good.
The believer knows that God (﷾) knows and decrees every event. That frees him from despair over what he loses, and from pride over what he gains.
The Prophet (ﷺ) said the believer is thankful in good times and patient in hard times. So everything that happens to him is good for him.
The Prophet (ﷺ) taught that true riches are contentment of the soul, not many possessions.
Most published studies link religious involvement with greater well-being, less depression and less suicide.
Cat Stevens was one of the best-known singers of the 1970s. He turned to God (﷾) after nearly drowning off Malibu in the mid-1970s. He became a Muslim in 1977 after his brother gave him a copy of the Qur'an. He then walked away from his pop career until 2006.