Short answer
No. In the same chapter Jesus says, "I ascend unto my Father, and your Father; and to my God, and your God". So Jesus had a God, the same God (﷾) as his disciples. Then John says why he wrote his whole book. He wrote so that readers would believe Jesus is "the Christ, the Son of God". He does not say they should believe Jesus is God. The Bible also uses the word god for other people. Moses was made "a god to Pharaoh", and the judges of Israel were called gods. Paul, a Christian missionary of the first century, keeps the two apart: there is "one God, the Father" and "one Lord Jesus Christ". The Qur'an records Jesus saying, "worship Allah, my Lord and your Lord". Islam teaches that Jesus was a prophet and a servant of God (﷾).
Full explanation
What Thomas said
Thomas was one of Jesus' disciples. He had refused to believe the others' report that Jesus was alive until he saw him. In John's account, Jesus then appeared and showed him his wounds. Thomas answered, "My Lord and my God." Christians see this as the high point of John's Gospel. A disciple calls Jesus God to his face. Jesus does not correct him, but blesses those who believe.
“And Thomas answered and said unto him, My Lord and my God.”
“Jesus saith unto him, Thomas, because thou hast seen me, thou hast believed: blessed are they that have not seen, and yet have believed.”
Earlier in the same chapter
Earlier in the same chapter, a week before Thomas spoke, John tells of Jesus meeting Mary Magdalene at the tomb. He sends her to the disciples with a message. He is going up to "my Father, and your Father", and "to my God, and your God". So even on the morning Christians call the resurrection, Jesus speaks of a God above him. It is the same God as his disciples' God (﷾).
“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.”
Why John says he wrote
Three verses after Thomas speaks, John states the purpose of his whole book. He wrote so that readers would believe that Jesus is "the Christ, the Son of God", and have life through his name. Whatever Thomas meant by his cry, John did not draw the conclusion Christians draw. He asks the reader to believe Jesus is the Christ, not that he is God. The Bible gives the title son of God to Adam and to Israel too.
“But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name.”
“Which was the son of Enos, which was the son of Seth, which was the son of Adam, which was the son of God.”
“And thou shalt say unto Pharaoh, Thus saith the Lord, Israel is my son, even my firstborn:”
One God (﷾), the Father; one Lord, Jesus
Paul, a Christian missionary of the first century, writes that there are many so-called gods and lords. But for Christians there is "one God, the Father", and "one Lord Jesus Christ". He gives the word God to the Father and the word Lord to Jesus. He keeps the two apart.
“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.”
"God" used of others in Jesus' time
The Bible uses the word god for more than the One God (﷾). In Hebrew the word is elohim, and in Greek it is theos. God (﷾) made Moses "a god to Pharaoh". The judges of Israel are called "gods". In Exodus 22:8-9 the King James Version translates the same Hebrew word, elohim, as judges. John does use theos of Jesus. But the Bible gives the same word to Moses and to the judges. Jesus himself quoted the psalm to answer a charge of blasphemy, which means speaking against God (﷾). Jewish writings from Jesus' own era use the word the same way. A Dead Sea Scroll, a Jewish text found near the Dead Sea, applies the elohim of Psalm 82:1 to Melchizedek, an ancient priest-king. Philo of Alexandria, a Jewish thinker of that time, says Moses was named god. A title given to Moses and the judges cannot, by itself, prove that anyone is the One God.
“And the Lord said unto Moses, See, I have made thee a god to Pharaoh: and Aaron thy brother shall be thy prophet.”
“I have said, Ye are gods; and all of you are children of the most High.”
“If the thief be not found, then the master of the house shall be brought unto the judges, to see whether he have put his hand unto his neighbour's goods. For all manner of trespass, whether it be for ox, for ass, for sheep, for raiment, or for any manner of lost thing, which another challengeth to be his, the cause of both parties shall come before the judges; and whom the judges shall condemn, he shall pay double unto his neighbour.”
“Jesus answered them, Is it not written in your law, I said, Ye are gods? If he called them gods, unto whom the word of God came, and the scripture cannot be broken; Say ye of him, whom the Father hath sanctified, and sent into the world, Thou blasphemest; because I said, I am the Son of God?”
“A Psalm of Asaph. God standeth in the congregation of the mighty; he judgeth among the gods.”
A Dead Sea Scroll from the first century BCE, found in Qumran Cave 11. It quotes Psalm 82:1, where the King James Version has "God standeth in the congregation of the mighty", and applies the Hebrew word elohim in that verse to Melchizedek, a heavenly figure who carries out God's judgement, not to the One God.
Philo, a Jewish writer in Alexandria in the time of Jesus, says that Moses was named god and king of the whole nation. He did not mean that Moses was the One God.
"My Lord and your Lord"
The Qur'an records the same message from Jesus that John 20:17 gives. He told the Children of Israel to worship Allah (﷾), "my Lord and your Lord". On the Day of Judgement, Jesus will say he told them nothing else. On this point the Gospel of John and the Qur'an agree. Jesus has a Lord, just as his followers do.
“They do blaspheme who say: "Allah is Christ the son of Mary." But said Christ: "O Children of Israel! worship Allah, my Lord and your Lord." Whoever joins other gods with Allah,- Allah will forbid him the garden, and the Fire will be his abode. There will for the wrong-doers be no one to help.”
“"Never said I to them aught except what Thou didst command me to say, to wit, 'worship Allah, my Lord and your Lord'; and I was a witness over them whilst I dwelt amongst them; when Thou didst take me up Thou wast the Watcher over them, and Thou art a witness to all things.”
Verse by verse
John 20:28 — Gospel of John, chapter 20, verse 28
“And Thomas answered and said unto him, My Lord and my God.”
Cited for
Thomas calls Jesus "my God", so a disciple confessed that Jesus is God.
The response
In the same chapter Jesus speaks of "my God, and your God". And the Bible gives the word god to Moses and to the judges of Israel without making them the One God.
Where it lands — Thomas's words, read in their own chapter, do not show that Jesus is the One God.
John 20:29 — Gospel of John, chapter 20, verse 29
“Jesus saith unto him, Thomas, because thou hast seen me, thou hast believed: blessed are they that have not seen, and yet have believed.”
Cited for
Jesus accepted Thomas's words and blessed those who believe the same, so he approved the belief that he is God.
The response
A week earlier Jesus had sent word that he was going "to my God, and your God". John tells us at once what belief he wrote to produce: that Jesus is "the Christ, the Son of God". That is the belief the Gospel of John asks its readers to hold.
Where it lands — The Gospel of John's own statement of purpose asks readers to believe Jesus is the Christ, the Son of God, not that he is God.
Perspectives
Mainstream Christian
Thomas calls Jesus his Lord and his God. Jesus accepts the words and blesses those who believe. John places this at the high point of his Gospel.
In John 20:28 Thomas says to Jesus, "My Lord and my God."
Rebuttal
John records the words, but in the same chapter Jesus speaks of "my God, and your God". The Bible gives the word god to Moses and the judges of Israel without making them the One God. Psalm 45 even speaks to a human king as "O God", and says his God anointed him.
Christians hold that by accepting the words and blessing believers, Jesus confirmed that he is God.
Rebuttal
Jesus had no need to correct him: a week earlier he had sent word that he was going "to my God, and your God". His reply praises believing without seeing. John then says the belief he wrote to produce: that Jesus is "the Christ, the Son of God".
The Islamic perspective
Thomas's words do not make Jesus God. In the same chapter Jesus speaks of "my God, and your God". John says he wrote so that people would believe Jesus is the Christ, the Son of God.
In the same chapter, on the morning Christians call the resurrection, Jesus calls the Father "my God, and your God".
John states his purpose: that readers believe Jesus is "the Christ, the Son of God".
John sums up the purpose of his book near its end. There he asks for belief that Jesus is the Christ, not that he is God.
Paul separates "one God, the Father" from "one Lord Jesus Christ".
The Bible calls Moses "a god to Pharaoh" and the judges "gods", and Jesus quoted the psalm himself.
Jewish writings of Jesus' era use elohim and theos of figures other than the One God (﷾). Examples are the Qumran Melchizedek scroll and Philo.
The Qur'an records Jesus telling his people to worship Allah (﷾), "my Lord and your Lord".