Everton made a late loan move for the Dutch forward on deadline day, offering around £10m to cover both the fee and his wages, after their pursuit of Folarin Balogun hit trouble over a failed medical.
Manchester United turned the approach down, with the transfer arriving too close to the deadline to allow time to bring in cover, meaning Zirkzee will now stay at Old Trafford until at least January.
The 25-year-old was one of several players United had made available for transfer over the summer after a difficult debut season in England, in which he scored twice in 24 Premier League appearances following his £36.5m move from Bologna.
He has featured only sparingly so far this term, with Matheus Cunha preferred through the middle and Benjamin Sesko's return to full fitness set to squeeze his opportunities further.
Carrick was nonetheless emphatic when asked about the striker's role heading into a season in which United will compete on four fronts.
"For sure Josh is important," he told reporters on Friday. "For the team, for the squad depth. Josh is happy here and I have a really good relationship with Josh.
"He wants to play more than he has but that is good healthy competition. We have a lot of games coming up, we have a shorter period between games, Josh can do some really top things, I have seen it myself."
The manager also gave his broader assessment of a transfer window that improved United's midfield but left question marks over the depth and quality of the defence.
"We went into the window wanting to come out stronger from the group we had last season, and we have definitely added to that — a lot of good work in that respect, and the situation is what it is in terms of finances and what we can work with," Carrick said.
"Certainly, the players we have brought into the group, absolutely delighted with — I think we have done some really good business. You can always look at it and think there are other things we could have done or might have done, but some things become available.
"Largely, I think we are in a good place with the squad, we have players that are flexible throughout the team and can play in different positions as and when needed."