A great deal depends on what you want to say. In general, you would reply to a question using the same construction as the question. So if asked, "Did you invite her to the party?", you would probably respond "I did not invite her," or simply, "I did not." And the same pattern would apply to "Have you invited her?.
However, there is a difference in the two phrases. "I did not invite her" refers to a specific point or period of time in the past, while "I have not invited her" refers to the present. So if asked, "Did you invite her to the party" and you respond, "I have not invited her", the change in tense from past to present would usually suggest that you might invite her in the future (*I have not invited her, but I'm thinking of doing so), or it might serve to emphatically (but implicitly) mean "I have not invited her (and I have no intention of doing so)," The difference would be established by tone, expression and body language.