I wholly agree with the elegant simplicity of the Gospel. So I hate the nitpick. But would you consider the resurrection an integral part of "believe on Jesus Christ"?
That is a great question. I think Acts 16:32 indicates that the jailer probably heard the entire story of Jesus.
Acts 16:32 And they spoke the word of the Lord to him together with all who were in his house.
Two verses later we read:
Acts 16:34 And he brought them into his house and set food before them, and rejoiced greatly, having believed in God with his whole household.
So, the jailer and those in his household heard the word from Paul and Silas. Given the fact that the resurrection is a central part of the gospel message, we can strongly infer that the jailer and his household were told about it.