The rules of Quoridor state that if a player has run out of walls, the player must move his or her pawn. However, is it a valid move for a player to choose not to place a wall (and therefore move), then choose to maintain the current position of the pawn? In other words, is remaining in place considered a valid move?
No, you must act on your turn. From the rules:
Each player in turn chooses to move his pawn or to put up one of his fences. When he has run out of fences, the player must move his pawn.
The second sentence there, which you reference in your question, is merely a clarification on the first: if you can't place a fence then you must move because you must always either move or place a fence.
The definition of "moving" your pawn is given in the rules:
The pawns are moved one square at a time...
...in particular, not zero squares at a time. This is also what is reflected in the plain English meaning of "move a pawn" (as opposed to "make a move in a game", which is a more abstract meaning of the word "move" that might include passing, i.e. doing nothing).
Note that the requirement to make a (non-empty) move every turn is a common feature to most if not all abstract strategy games. Chess even invented a name for the situation where any move you make will be bad for you: zugzwang.