The short answer is ‘No’. In Islam the decision of who goes to heaven and who goes to hell is left entirely to God as He alone knows people’s hearts and is aware of their deeds. What Islam claims is that it is the perfect religion for mankind and a religion for all time and all people.
This day have I perfected your religion for you and completed My favour upon you and have chosen for you Islam as religion. (Ch5. V.3)
It therefore certainly claims to offer the best guidance – which if followed will lead to paradise – but simply being a Muslim is not enough to enter paradise. It is the righteous who are rewarded by God, who may well be Muslims, Christians, Jews etc.
The Qur’an states that people who do good deeds will be rewarded for them:
For those who do good deeds, there shall be the best reward and yet more blessings. (Ch.10: V.27)
So it leaves it open to God as to who will be judged worthy of entering paradise. Islam also tells us the qualities of the people who will enter paradise:
Surely, those, who believe and do good deeds, and observe Prayer and pay the Zakat, shall have their reward from their Lord, and no fear shall come on them, nor shall they grieve. (Ch.2: V.278)
And if they had believed and acted righteously, better surely, would have been their reward from Allah, had they but known! (Ch.2: V.104)
In the above two examples, those who believe in God, do good deeds, act righteously, observe prayer and give to charity are promised to be rewarded by God – and this may include the ultimate reward of being admitted to Paradise in the Hereafter.
Surely the believers and the Jews and the Christians and the Sabians – whichever party from among these truly believes in Allah and the last day and does good deeds – shall have their reward with their Lord and no fear shall come upon them, nor shall they grieve. (Ch.2: V.63)
If a person rejects Islam after knowing Islam and fully understanding its truth he will be asked about it by God. Otherwise he will be judged according to his own religion or his understanding of right and wrong.