Short answer
No. The verse says "form", and form is not the same as being God. The Greek word, morphe, can mean an outward look. Mark 16:12, a passage added later to the end of Mark, uses it when Jesus appears "in another form" to two followers. Many scholars read the verse against Genesis, where Adam and Eve reach for the serpent's promise to be "as gods". On that reading, Jesus is the one who did not reach for equality with God (﷾). The next verses say that God (﷾) lifted him high and gave him the name above every name, all to the glory of God the Father. Paul, a Christian missionary of the first century, also writes that the head of Christ is God. Jesus said, "my Father is greater than I". So the passage shows a servant honoured by God (﷾). Islam teaches the same: Jesus was a servant of God (﷾).
Full explanation
How Christians read the verse
Christians read "in the form of God" as meaning that Jesus shared God's (﷾) very nature before he was born. They point to the matching phrase "the form of a servant". He truly became a servant, they say, so he was truly God. They also read the verse as teaching that he laid aside his rights when he became a man.
“who, existing in the form of God, didn’t consider equality with God a thing to be grasped, but emptied himself, taking the form of a servant, being made in the likeness of men.”
What the word morphe means
Everything turns on the Greek word morphe (form). J. H. Thayer, a nineteenth-century scholar, wrote the standard lexicon (word-book) of New Testament Greek. He defines morphe as the form by which a person or thing strikes the eye: its outward appearance. His example is Mark 16:12, where Jesus appears to two followers "in another form". That verse is from the longer ending of Mark, a later addition, but it is still early Christian Greek. Nobody reads it as saying he changed his nature. So the word alone cannot prove that Jesus is God.
Defines morphe (form) as the form by which a person or thing strikes the eye, its external appearance. It gives Mark 16:12, where the risen Jesus appears "in another form", as an example.
“After that he appeared in another form unto two of them, as they walked, and went into the country.”
Not grasping at equality
Next, look at what the verse says Jesus did not do. He did not treat equality with God (﷾) as something to seize. James Dunn, a modern scholar of the New Testament, and others read this against the story of Adam. In Genesis, the serpent promises Adam and Eve that they will be "as gods", and they reach for it. Jesus, the obedient servant, did not reach for equality with God (﷾). A person does not reach for what he already holds.
“who, existing in the form of God, didn’t consider equality with God a thing to be grasped, but emptied himself, taking the form of a servant, being made in the likeness of men.”
Argues that Philippians 2:6-11 is best read against the story of Adam: where Adam grasped at being like God, Christ did not. On this reading the hymn need not teach that Christ existed as God before his birth.
“For God doth know that in the day ye eat thereof, then your eyes shall be opened, and ye shall be as gods, knowing good and evil.”
What the passage says next
Paul, a Christian missionary of the first century, does not stop at verse 7. He says that God (﷾) exalted Jesus and gave him the highest name. Every confession of Jesus as Lord, he says, is "to the glory of God the Father". The one who exalts and the one who is exalted are two. In another letter Paul states it plainly: for Christians there is one God, the Father, and one Lord, Jesus Christ.
“Therefore God also highly exalted him, and gave to him the name which is above every name, that at the name of Jesus every knee should bow, of those in heaven, those on earth, and those under the earth, and that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.”
“But to us there is but one God, the Father, of whom are all things, and we in him; and one Lord Jesus Christ, by whom are all things, and we by him.”
Paul and Jesus place the Father above the Son
Elsewhere Paul says that the head of Christ is God. He says that at the end the Son himself will be made subject to God (﷾), who put all things under him. Jesus says the Father is greater than he is. John says that after he rose he called the Father "my God, and your God". God (﷾) has no head over him, nothing greater than him and no God above him.
“But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.”
“For he hath put all things under his feet. But when he saith all things are put under him, it is manifest that he is excepted, which did put all things under him. And when all things shall be subdued unto him, then shall the Son also himself be subject unto him that put all things under him, that God may be all in all.”
“Ye have heard how I said unto you, I go away, and come again unto you. If ye loved me, ye would rejoice, because I said, I go unto the Father: for my Father is greater than I.”
“Jesus saith unto her, Touch me not; for I am not yet ascended to my Father: but go to my brethren, and say unto them, I ascend unto my Father, and your Father; and to my God, and your God.”
What the Qur'an says about Jesus
The Qur'an records Jesus' first words as a servant of God (﷾), given revelation and made a prophet. This fits the humble servant Paul describes in Philippians 2, not a God in human form. Islam teaches the same: Jesus was a servant of God (﷾).
“He said: "I am indeed a servant of Allah: He hath given me revelation and made me a prophet;”
Verse by verse
Philippians 2:6-7 — Epistle to the Philippians, chapter 2, verses 6-7
“who, existing in the form of God, didn’t consider equality with God a thing to be grasped, but emptied himself, taking the form of a servant, being made in the likeness of men.”
Cited for
"The form of a servant" means Jesus was truly a servant, so "the form of God" must mean he was truly God.
The response
The parallel does not prove what it is meant to prove. Form is how one appears, as in Mark 16:12. The word does not by itself say what one is. The first meaning Thayer's lexicon gives for morphe is "the external appearance", and Mark 16:12 uses the word in exactly that sense. In the same passage God (﷾) exalts the servant and gives him his name, which no one says of God (﷾).
Where it lands — The word "form" cannot carry the weight of "God". Paul himself names the one God (﷾) as the Father.
Philippians 2:6-7 — Epistle to the Philippians, chapter 2, verses 6-7
“who, existing in the form of God, didn’t consider equality with God a thing to be grasped, but emptied himself, taking the form of a servant, being made in the likeness of men.”
Cited for
Jesus already had equality with God and chose not to use it for his own gain.
The response
The verse does not state that he had equality. On the Adam reading, it says he did not reach for it. The King James wording, "thought it not robbery to be equal with God", is an older rendering. Two later Bibles render the Greek word harpagmos as "a prize" and "a thing to be grasped". They are the Revised Version of 1881 and the American Standard Version of 1901. The World English Bible (WEB) follows them. Elsewhere Paul teaches that God (﷾) is the head of Christ and that the Son will be subject to God (﷾). Jesus said the Father is greater than he.
Where it lands — Paul and Jesus both place the Son below the Father, so this verse cannot mean the Son is equal to God (﷾).
Isaiah 45:23 — Nevi'im, Yeshayahu, chapter 45, verse 23
“I have sworn by myself, the word is gone out of my mouth in righteousness, and shall not return, That unto me every knee shall bow, every tongue shall swear.”
Cited for
Verses 10-11 apply Isaiah 45:23 to Jesus. There the LORD says "unto me every knee shall bow". So, the objection runs, Jesus receives the bow owed to the LORD.
The response
In the book of Isaiah the bow belongs to the LORD. Paul keeps it so: knees bow at the name God gave Jesus, "to the glory of God the Father". The glory passes on to God (﷾); it does not stop at Jesus. In another letter Paul says that for Christians there is one God, the Father, and one Lord, Jesus Christ.
Where it lands — The honour is a gift from God the Father, and it ends in his glory, so it does not make the one honoured God.
Perspectives
Mainstream Christian
Philippians 2:6-11 teaches that the Son existed as God before his birth and humbled himself to become a man. The parallel with "the form of a servant" shows that "the form of God" means the divine nature.
"Being in the form of God" means that Christ shared the divine nature before he became man.
Rebuttal
Christians may read "form" as nature. But the first meaning Thayer's lexicon gives is outward appearance, as in Mark 16:12. The same passage says God (﷾) exalted Jesus and gave him his name, which no one says of God (﷾).
"The form of a servant" means he was truly a servant. So, they say, "the form of God" must mean he was truly God.
Rebuttal
Grant that Jesus truly became a servant. Even if "form" meant what one truly is in both lines, the passage ends with God (﷾) exalting the servant. That honour is to the glory of God the Father, and the one honoured is not the one who honours.
The Islamic perspective
Jesus was a servant and messenger of God (﷾). Philippians 2 speaks of his form, his humility and his reward from God (﷾). It never says he is God, and it ends by giving the glory to God the Father.
The Greek word morphe can mean outward appearance, as in Mark 16:12 (in the longer ending of Mark). So "form of God" does not by itself mean "God".
Take the Adam reading of the verse. It says Jesus did not grasp at equality with God (﷾), as Adam and Eve do in Genesis.
In the same passage God (﷾) exalts Jesus and gives him his name, to the glory of God the Father.
Paul says there is one God, the Father, and one Lord, Jesus Christ.
Paul says God (﷾) is the head of Christ and the Son will be made subject to God (﷾).
Jesus says the Father is greater than he is and calls the Father his God.
The Qur'an teaches that Jesus was a servant of Allah (﷾) and a prophet, not God.