him staring

Yes it is correct. Possibly a little casual and conversational, but not howlingly incorrect.

I am aware that in US English, in particular, In it is usual to use the possessive case to modify a gerund, so that "I don't like his staring at you" would be preferred.

Traditional English grammar distinguished between gerunds and present participles. Both terms refer to the non-finite verb form ending in -ing (standing, swimming, etc.); traditionally, the former was applied when the verb form was acting in some sense like a noun (say, as the subject or subject of a verb or preposition), and the latter when it was acting in some sense like an adjective. The analogous distinction is very clear in Latin, where gerunds and participles are declined as nouns or adjectives, but the line is blurrier in English, and many modern linguists reject this distinction. The Cambridge Grammar of the English Language, a widely respected descriptive grammar by Rodney Huddleston and Geoffrey Pullum, uses the term gerund-participle, and lists its various uses without commenting on which might be considered nominal and which adjectival.

These are all pefectly acceptable in normal BrE:-

I don't like him staring at you

I don't enjoy you shouting at me

I don't remember him mentioning it.

Many thanks, Contrex, for your detailed reply.

Best wishes

what about,

I don't like him being staring at you.

No, definitely not. You can only have one gerund/participle. Anyway, "being staring" is nonsense.