Steven Lawless has been given a new deal by Partick, a few days after being handed a ban for breaching betting rules.
Midfielder Miguel Pallardo has signed a new one-year deal with newly-promoted Hearts.
McCullough, who has represented Northern Ireland at U17 and U19 level, had been on trial with Paul Dickov's side at the Keepmoat Stadium since the start of the pre-season.
He signed for United's Academy aged of 16 before going on to captain the youth side throughout the 2011/12 campaign.
The 19-year-old told Rovers' official website: "I heard Doncaster were interested in taking a look at me so I came across and I've not looked back since - I've really enjoyed every minute that I have been here.
"Pre-season has been tough but as the manager has said it will give us a good base to build upon going into the season.
"The manager seems to have a good plan and wants the boys to keep improving.
"The rest of the lads have been very friendly and have made me feel very welcome - there's a good team spirit here."