In once of my recent live games, I reached a winning position, Q vs. R with no pawns on the board. However, in reaching that position, I used much more time than my opponent, so that I only had about 2 minutes left to convert the win. With just seconds remaining on my clock I offered a draw but my opponent refused it, and when I lost on time he refused my rematch request. Is that bad sportsmanship on his part?

Well, with the time and draw thing; everyone wants to win in chess. When you had a few seconds on your clock he knew he was going to win, why would he want to accept a draw when he knew that. I would of done the same if I was in a lost position. Its just the game of Chess, its ok.

To the OP, the short answer is: No

Yeah, there's nothing wrong with taking advantage of your opponent's time trouble in a rapid time control game of chess, and no one is obligated to a rematch.

Moreover, drawing a QvR endgame for the side with the Queen should have been a fairly easy and straight forward task, even with only 2 mins on the clock.

No, it isn't bad sportsmanship on his part.  And no one is owed a rematch automatically, maybe he has something else to do, or just realized you will beat him and doesn't want it.  You can't know, so let it go.

But it IS dumb play on your part.  With K+Q v K+R, if you want a draw, you don't ask.  You capture the Rook.