No. Medical records should not be accessible to background checks for guns, HOWEVER...
There should be a list (like the terrorist watch list) except this might be the firearms mentally challenged list whereby if you are deemed by a medical professional/board/committee to be a danger to yourself or others by use of ANY weapon, then you should be on that list of people who should not be able to obtain a firearm. It's sad to say this, but it is a necessary course in ridding the gun society of people who continue to pose a threat to our rights.
By the same token, those same people probably shouldn't be able to drive a car or operate an machinery that could cause harm to themselves or others. This is not just for guns, but for the overall safety of mankind in a civilized society.