Steve,
Chess.com is like many other internet sites where people can chat and post: some people use the relative anonymity to vent some personal negativity. This "rude to not resign" or "rude to ask someone to resign" thread should probably be in the FAQ by now.
Most of the responses to your post have rightly pointed out that your opponent's method of asking you to resign was thoroughly rude. It is also true that playing a completely lost position is considered rude by many, though not all. PawnFork's post (#23 on this thread) seemed a reasonable compromise. The game is not completely lost to you unless you yourself see exactly how you are going to be checkmated. It is reasonable for an experienced player to explain some chess culture to a beginner, but only in a completely civil, friendly, and welcoming way. I would recommend:
"Hey Steve, I've been enjoying this game. I believe I have a forcing line with 24...Nf6 25.Kf8 Qa8 26.Kg7 Ne6. Note that my g pawn protects my f6 Knight. I thought you might be a beginner, because in most chess communities it is considered etiquette to resign once the mate is inevitable - so I thought I'd mention it. It doesn't put me out to finish the game, but I thought I'd fill you in on some chess culture. One reason for the etiquette of resigning is in a situation where the game is the last one in a group for a tournament, and the winners of all the other groups are waiting for this game to end so that the next round of the tournament can begin. But it's your call; I'm happy to finish this with you."
I hope that helps. About 95% of the time we have this thread, it's because the dude who wants someone else to resign is just a crank - but it is true that resignation is considered polite for completely lost situations. But do wait until its completely lost. Many people are so used to resigning or having their opponents resign that they haven't practiced their endgame. The golf analogy was great: if everyone gave up after one of Tiger Woods' incredible drives, no one would ever know if he could putt well enough to close the deal! Endgames are an exciting part of the game. Openings are all about memorization, but endgames are very personal conflicts. When you have that many hours or days invested in a game; the anxiety, pressure, excitement, and emotional investment really picks up.
I draw an analogy to golf here: to me, expecting someone to resign is the same as expecting them to concede every two-foot putt for 17 holes, and then when they get to the 18th, and the match is on the line, forcing you to make it by NOT conceding it, at which point, if you are never forced to MAKE that putt, two feet looks like 20 because you have not made that simple putt all day. I would prefer to finish games whenever possible to stay sharp, and I certainly do not find an opponent who will not resign offensive, for perhaps he simply wonders if I am CAPABLE of "making that two-foot putt" or he wants to learn the correct stroke so that when he is faced with that "putt" himself he will know how to "strike" it.
Yeah I like spending a generous amount of time at the 19th too!!