The father should be sacked from his role as a doctor because a fundamental of being a doctor of any kind is 'first, to do no harm', non-maleficence, a basic of Hippocratic ethics.
Instead he has knowingly radicalised his son to be a hate-filled murderous bigot, as his own social media posts openly demonstrate.
Regardless of which 'side' he is on, he is not fit to be employed by the NHS or anyone else.