Earlier in my game I had moved my king 1 space to right of its original position. Later in the game, after my opponent castled he placed his rook correctly which resulted in the rook being directly across from my king on the other end of the board with no pieces in between. He then stated I was in check. Is this a legal check since he actually moved his king first in the castling action, or should he only be able to call a check after my next move? In other words, can he check me from a castling move since his king did not originally have me in check?
Yes, of course he can; you're indeed in check. Imagine if checks didn't exist--then on the next move, if your king hasn't moved, he could just take your king with his rook.
Yes, it is check. It doesn't matter which piece was moved.
[Fen "8/8/8/8/8/8/8/R3K1k1 w KQkq - 0 1"]
You can also have discovered check where your opponent moves a piece, exposing a piece behind it which attacks your king. This is another example of a check in which the checking piece wasn't moved.
[Fen "4r3/8/4k3/8/8/4K3/8/8 w KQkq - 0 1"]
[Fen "8/4pr2/5k2/3b1P2/8/4B3/4KR2/8 w KQkq - 0 1"]