Ingore the guy! That is probably the best compliment an honest chess player can recieve so take it as such and continue taking it to the guy! Good luck.
Cheating

Disable chat, block your opponent, finish the game, and move on with your life. OR if you're really upset, then take a screenshot of the chat and send in with a "ticket".
http://support.chess.com/Tickets/Submit
Let me see if I can find the official word:

People seem to do that because it's easy, they know it's offensive and it bandages their bruised ego. In reality it says more about them than about their opponent. If it upsets you, he wins; if you laugh it off, he loses again.

Jabbok sometimes uses hints program )

People seem to do that because it's easy, they know it's offensive and it bandages their bruised ego. In reality it says more about them than about their opponent. If it upsets you, he wins; if you laugh it off, he loses again.
Isn't that cheating though?

Jabbok sometimes uses hints program )
I think you ought to explain that comment, or retract it.

Jabbok sometimes uses hints program )
What is hint's program?
Also, as a side note, I got accused of using an engine in bullet chess once. How is that even possible? The only way I know that people cheat is to enter their moves into an engine but that would take time. Clearly if someone tried to do that in bullet chess they would lose on time every game.

Jabbok sometimes uses hints program )
What is hint's program?
Also, as a side note, I got accused of using an engine in bullet chess once. How is that even possible? The only way I know that people cheat is to enter their moves into an engine but that would take time. Clearly if someone tried to do that in bullet chess they would lose on time every game.
Yeah, it's possible.
I'm not going to explain how to do it, for obvious reasons.

Jabbok sometimes uses hints program )
I think you ought to explain that comment, or retract it.
We play in the tournament <2000. I think running at two or three moves ahead and my rating in 1800 (live, online, offline), and I just counted five moves ahead, he believes strongly in 2000, I think it is not without cheating. See max ratings.
Some guy that I'm playing has sent me a chat accusing me of cheating (with some choice explatives!)
I'm offended by this accusation and tempted to resign the game but I don't want to give him the satisfaction.
I recently fell from a 1600 ranking down to the 1400's but it was because I was hospitalized and not been feeling well. I haven't paid very close attention to my games and my rank suffered for it. Now I'm at home and trying to climb back up but I don't want this fluctuation in my rank to be grounds for accusations.
What's the best course of action for me to take?