can I put my opponent in checkmate in the move following his putting me in check?

Help! He's calling it either a draw or a stalemate. I say I won!

If you are in Check

You can not do anything until you are out of check.

Since I'm usually wrong I probably misunderstood your thread header, but otherwise that is the rule.

You can checkmate your opponent on your turn even if you are currently in check. All that is required is that your move reaches a checkmate position while simultaneously relieving the check on your king.

key words " relieving the check on your king"

Yes, it's possible to answer a check with a checkmate. It's very rare tho and is usually only seen in composed problems.

simplest position

Or maybe something of a more familiar nature - discovered check

http://chess-problems-gr.blogspot.gr/2008/04/iatridis-stavros.html

This is a famous chess problem (mate in three).

In the main line 1.Rh4 Qg4, white plays 2.Kxb6!! and exposes his king to EIGHT different checks. He replies to all eight of them with checkmate.

I think it's about who's king is off the board first. Let's say you put a king in check then your opponent lays checkmate. But since theoretically you didn't get your king to safety you would take your opponent's king then your opponent would take your king. Also I think if you are in check you are required to either block or move the king and I think if the block you make does checkmate then it is legal.

Wfloh put simply if the move you make To get out of check also checkmates them then you're fine.

I got checkmate but it said stalemate??? Is it because one of my opponent's moves could have put me in check???

If it says stalemate, then your opponent is not in check, it's their move, and they have no legal moves.

Wow, you don't know how to chess?

Was your computer hot, cause if so it might be that chess.com miscalculated