I think the reason Young hid the footage of Rush taking the gun is that if the civilians and military personnel knew Rush would frame people for murder, morale would really go down the tubes.
Young couldn't have been planning to leave Rush on the planet because they hadn't encountered it yet, and couldn't know they'd ever encounter a planet with a wrecked space ship that would both interest Rush and cause Rush to want to stay after the rest of the team heading back toward the gate.
Rush could certainly be successfully prosecuted for his actions regarding the gun, specifically tampering with evidence, perjury, etc.
In contrast, Young couldn't be successfully prosecuted for stranding Rush. He can't be forced to testify against himself, and there were no other witnesses since Rush was unconscious. Rush could try and make something up, which is just he said/she said, and Rush's testimony would already be compromised because of his actions regarding the gun, which shows he is perfectly willing to lie and frame Colonel Young. So his testimony would be disregarded.
As I recall the episode, the rest of the team had to leave to get back to the gate in time, and Young and Rush were going to be pushing the limits and cutting things close before their fight. Since Rush was unconscious until long after Destiny had departed, Colonel Young would've had to bodily carry him all the way back to the gate, and we know Colonel Young was in very bad shape from the fight. It's quite probably that he couldn't have physically made it back in time had he tried, in his weakened condition, to carry or drag Dr. Rush to the gate. Nothing in civil or military law says you have to sacrifice your own life to try and save someone else, and we actually don't know that he didn't try and fail to pick Dr. Rush up.
So really, nobody could make a successful case against Colonel Young, except perhaps a military court that might find he acted without good order and discipline. You could certainly charge Colonel Young with assault and battery, but again, you can't prove who threw the first punch unless Colonel Young is willing to testify against himself, and Rush's testimony can't be trusted because he'd already tried to frame Colonel Rush for murder. On top of that there's a clear case against Dr. Rush for hijacking and mutiny.
So we're just left with rumors and discontent about Young amongst the civilians, but nothing that would provide a legal basis for removing him from command except a judgement by his military superiors that he has an inability to lead.