I have my rook on the 8th rank. My pawn is directly behind it. I want to move diagonal with my pawn to promote it to a queen. Can I move diagonal to reach the 8th rank and promote my pawn?
You may do so only if the pawn is capturing one of the opponent's pieces; that is the only way a pawn ever moves diagonally. So here you can do what you describe:
[fen "3Rn3/k2P4/8/8/8/K7/8/8 w - - 0 1"] 1.dxe8=Q
But if you're talking about a situation like the following, then no, the pawn cannot move as you described:
[fen "3R4/k2P4/8/8/8/K7/8/8 w - - 0 1"]