Mourinho on Friday described Wenger as "a specialist in failure" - referring to Arsenal's eight-year trophy drought - after Wenger implied the Premier League title was Chelsea's to lose.
Wenger insisted his comments saying any team which says they are not in the title race - as Mourinho has done countless times this season - does so out of fear of falling short of their ambition.
The Frenchman claimed his comments were not aimed at Mourinho and slammed his rival for his controversial riposte, and claimed the Portuguese had brought "embarrassment" on Chelsea.
"I do not want to go into that silly, disrespectful remark," Wenger told BT Sport following his side's FA Cup fifth-round 2-1 win over Liverpool.
"I never spoke about him in my press conference and I will not start.
"The only thing I know is it's more embarrassing for Chelsea than for me."
Wenger and Mourinho endured a fractious relationship during the Portuguese's first spell and a fragile truce was eroded on Friday.
"He is a specialist in failure. I'm not," said Mourinho, whose side lost 2-0 to Manchester City in the FA Cup fifth round on Saturday.
"If supposedly he is right and I'm afraid to fail, it's because I haven't failed many times."
Former Arsenal goalkeeper Bob Wilson on Sunday morning told BBC Five Live's Sportsweek programme Mourinho's remarks were "disrespectful" and "despicable".
Wenger was reluctant to return fire.
"I'm embarrassed for him, honestly," Wenger said. "I'm more disappointed for Chelsea than for me.
"I'm not interested in the subject at all. That is really genuine.
"If you are interested and you have nothing better to do in your life, you are free to do it.
"Honestly I cannot force my interest in things that are not interesting for me.
"I love football - and I'm sorry to disappoint you - what's interesting for me is what happens on the pitch."