Homer didn’t deny anything, he wasn’t even involved in that decision. He even put in a good word for her. However she basically lied when she was asked about it and that dug her in deeper, also leaving the post up didn’t help.

NASA basically wanted her to not draw controversy to the agency over social media, per their rules, and they normally have a reasonable amount of leeway. But there were a couple concerning things during the process, mistakes she made (that she has probably since grown from) that led to NASA passing on her internship.