Rose for a few seconds wanted to die with Jack. In the film after she realized that he was dead she grabbed his hand and rested her head on it while the lifeboat sailed by them.

Then she remembered what she promised him, and then got up and started to yell, or try to yell, "come back".

And if I recall correctly, God says in the Bible that it's ok to re-marry if your wife or husband dies. But otherwise, it would be sin-ful. IIRC