A GP has as much a duty of confidentiality as a solicitor has.
There is No Way any GP would investigate anyone - that is 100% not their business. To be honest, you asking that would probably flag you up in their mind as having possible mental health issues yourself.
(Apart from which there's not a GP in the UK who's not already overworked trying to keep the NHS alive and their patients seen!)
Have you tried getting Citizens Advice to help you? They usually have a solicitor who volunteers his/her service free one day a week, and will give advice - in my case, also going so far as to send the first "legal warning" letter for me, which did the trick.
As PP's say, it's not - sadly - illegal to be nasty to people. However if the nastiness results in putting the person being abused into "a state of fear or alarm" that IS (at least in Scotland) an offence and the police do have a duty to intervene in such a case.
I just looked up how England deals with this (sorry I haven't time to look up Wales and N.Ireland!) and if you live there, here's what you need to know:
Section 2 of the Protection from Harassment Act 1997 as amended (PHA), and 'putting people in fear of violence' offences under section 4 of the PHA. The term can also include harassment by two or more defendants against an individual or harassment against more than one victim.
A prosecution under section 2 or 4 requires proof of harassment. In addition, there must be evidence to prove the conduct was targeted at an individual, was calculated to alarm or cause him/her distress, and was oppressive and unreasonable.
Next time you speak to the police tell them you expect them to act, based on this legislation.
Good luck