This is where the concept of fake news began ten years ago. Fox does this kind of thing weekly. I think something similar is happening in DC. They've invented stories using fake video, and the president believes it because it's on TV. So do people on his team. So no, you won't be seeing any resignations, and you also won't be seeing any FCC investigations. Of course, it could be said that the FCC doesn't regulate cable news, but there is no oversight of what happens there. The president complains about what he calls MSDNC, but Fox News consistently invents stories that are designed to promote the president's agenda. Nobody will get fired for it. Meanwhile, as you say, everyone else has to play under very different rules.
I read a lot of criticism of Norah O'Donnel's interview with the president. People felt she should have argued with him, and corrected all his mistakes. He does interviews with Fox News almost weekly. He just did one with Laura Ingraham. No hard questions. All cream puffs. It was a love fest. Nobody complained, because that's what they do. By now it's expected.