All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Wearing gold is permissible for women but forbidden for men. Hence, there is nothing wrong in repairing women's gold watches.
As for men's watches, if there is a tiny amount of gold in them, such as the hands of the watch, and not the entire watch that is made of gold, then there is also no harm for a man to wear it according to the view of many jurists.
Ibn ‘Uthaymeen was asked in Sharh al-Mumti’:
If a person says that the watch is not gold and it is not gold-plated, but in its machinery, there is some gold; is it permissible (to wear it)?
Answer: “Yes, there is nothing wrong in it because if it is in the interior machinery, then no one sees it or knows about it. If it is in the exterior machinery, such as the hand [of the watch] for example, then it is intended in itself (it is not bought for the gold in the hand), so there is no harm in using it.” [End of quote]
If the men's watch is entirely or mostly gold, then it is not permissible to repair it because reparing it is helping in evil.
The Fatwa of the Standing Committee reads:
“It is not permissible to repair the machines [equipment] used in what is forbidden because this is helping in falsehood.” [End of quote]
In this case, the money earned from repairing a watch that is entirely or mostly gold is forbidden, but not all your father’s earnings are forbidden. Indeed, there is no harm in benefiting from the earnings of someone whose money is mostly lawful but he has some ill-gotten earnings in them.
Allah knows best.