According to the Bible, one must accept the forgiveness that Christ extended when he died on the cross in order to be saved.
I assume you subscribe to the notion that salvation is a prerequisite for some sort of eternal life. I do this because that idea is prevalent so excuse me if I am in error. Also, I must quote the following:
John 17:2-3 "For you granted him authority over all people that he might give eternal life to all those you have given him. Now this is eternal life: that they may know you, the only true God, and Jesus Christ, whom you have sent."
The belief held over the falsity or veridicality of an assertion is not in itself something sufficient and I will claim necessary for the acquisition of eternal life. Knowledge seems to be required. What sort of knowledge? The Greek word is ginosko, implying not a belief or a proposition but an immanent knowledge of that.
The belief held is not sufficient in itself and is a preliminary stage of faith. Knowledge of the sort meant there is required. Of course that is not to say that that is all eternal life is but it certainly meets sufficiency conditions.
So to answer your question more directly, given the existence of salvation, a belief in a Jesus or His forgiveness is not necessary for eternal life and since the salvation is often seen as one step in the grand plan or scheme of things of achieving eternal life. What does seem to be necessary-and-sufficient is knowledge, but of a paricular kind.
Cheers ! 🙂