Ask for a rematch if you want another game. I don't think it matters if you win or lose.
etiquette question

It is in bad taste not to grant a re-match if the opponent wants one, and did not provide a reason not to grant one (such as saying something rude).

You can ask the person losing if they want a rematch. This gives them the chance to try it again. It's really important to people who lost in an extremely close game. Finally, you should always accept a rematch from the person that lost. Even if you won against a highly rated player, take the rematch. I won a person whose rating was over 300 points higher than mine. I accepted the rematch after winning because it was the right thing to do.

I generally ask only if I lost. There isn't a custom though, so just do what you feel comfortable with.

I think the etiquette question should be: Is it proper chess etiquette for the winner to GIVE a rematch if challenged?

Happy Thanksgiving to everyone stateside.
Thanks for the input. I had a close game and my opponent lost on a blunder.
I waited for them to ask for a rematch which was well deserved, but
there was no request. I thought to send a rematch request but thought it
might be rude. With being new to the site I felt it better to ask if there is
a status quo.

Send him a trophy and suggest that you are always ready for another, when desired!
HAPPY THANKSGIVING!
Depends - if u cant (like say - pressed for time) , just say so and leave - dont see anything wrong with that

i never accept rematches in blitz chess b/c I would ratherhave my opponents not play against my style. However, in any longer game formats I almost always accept rematches

requesting a rematch is fine whether you won or lost. before i ended up with too many games, i used to request a rematch evertime i lost.
i don't think it is bad etiquette for somebody to refuse; they may not have time, or simply not see the worth if they have beaten you convincingly.
sometimes if you offer a rematch when you're winning, it reduces the number of moves an opponent makes hopelessly continuing. also less chance of them disappearing and making just minimum moves, i.e. 1 move per 3 days, until the bitter end.

Send him a trophy and suggest that you are always ready for another, when desired!
HAPPY THANKSGIVING!
I think this is the best option. You should always give your opponent a trophy at the end of a game (that didn't time out) win or lose (IMHO). If you won the game, let your opponent know that you would be willing to play a re-match, but let the losing player actually issue the challenge when they're ready (maybe they want to train a little more first to prepare for it).

After a game is it customary for the winner or loser to suggest a rematch?
And is it in bad taste to ask for a rematch if you won?
Etiquette is a personal matter - manners if you like. The same standard does not apply to all, it's truly up to you.
It is in bad taste not to grant a re-match if the opponent wants one, and did not provide a reason not to grant one (such as saying something rude).
I don't think there is any obligation to accept rematch and you don't have to provide any reason. There are no rematches in the real OTB chess (except rarely on the highest level possible), so get used to it. You play in the tournament, usually you play with the same person once, so do your best. You can meet at another tournament, here is your chance. For example I once lost in active tourney to one guy, then in 2 weeks played with him in regular one. I really wanted to get revenge, worked on the opening ( I knew I will be playing him with the same color half a day before and actually we played the same opening), tried not to give him any chances ( he won by finding combination in the worse position ) and won.
Saying that, I mostly accepted challenges playing online blitz. Sometimes it was very annoying when after winning a tough game with putting max effort in it the guy asked for rematch right away. Like he knew that I will be tired, will play worse and lose. Later I started to refuse such rematches ( just disconnected, then connected again). With correspondence games here I had only one rematch, the guy played unusually bad for his rating and lost in 12 moves. So, I accepted the rematch and lost in about 40 moves, he played much better. He later explained that he was distracted by his grandchild. So, it was completely right to accept rematch.
Proper etiquette for me means you behave like you would behave playing face-to-face, i.e. being polite, not offering draw more than once, not commenting on the game in any way, etc. That's it. The chess laws say the same.
After a game is it customary for the winner or loser to suggest a rematch?
And is it in bad taste to ask for a rematch if you won?