People rightly laughed at Trump when he claimed this in the 3rd debate this past week. The thing is, I think the audience was laughing at the notion that Trump could respect women, when the real problem is what Trump thinks a woman is.
There are multiple definitions of respect, none of which strike me as problematic (these are the verbs; the noun versions are similar):
: to feel admiration for (someone or something) : to regard (someone or something) as being worthy of admiration because of good qualities
: to act in a way which shows that you are aware of (someone’s rights, wishes, etc.)
: to treat or deal with (something that is good or valuable) in a proper way
It is certainly plausible that Trump’s feelings towards women match the first and third definitions of respect here, provided we accept some rather retrograde definitions of the good qualities of women and what proper treatment of women entails. I don’t think it would be terribly unexpected if Trump’s definition of “woman” meant that groping and grabbing were “treating and dealing with something that is good or valuable in a proper way”. Likewise, regarding a woman not as a fellow homo sapien with mostly aligned priorities but as an unfathomably attractive but ultimately aesthetic object probably only deepens his offense at people lacking womanly qualities calling themselves that. Metal fans aren’t just indifferent but morally revolted when Limp Bizkit fans (such as they exist) call their idols metal.
There are hints of this problem in the responses of Trump’s Republican (erstwhile) allies as well: Paul Ryan insisted that “Women are to be championed and revered, not objectified”, which certainly doesn’t sound like an attitude one takes towards people essentially like oneself. Whether or not Ryan respects women isn’t the issue; it’s clear that he does. He just thinks of a woman as something like a rare giant catfish or a goldtop Gibson Les Paul.
Living in Japan exposes you to this attitude a lot: People don’t think they’re being sexist in a disrespectful sense by having only men drive on the freeway or having women do all the housework. They’re being completely respectful of what they think are the essential qualities of men and women. A lack of respect isn’t the issue: if anything too much respect is exactly the problem.