Can the COC deny paternity leave of 21 days for a secondary caregiver?

Reading Army Directive 2019-05, I noted that it states, more than once, that the commander "may approve." When trying to interpret regulations and policies, attention must be paid to whether it states "may" versus "will," or "shall."
Yes, context must be considered, but generally, "may" means a choice to act, while "will," or "shall," makes it imperative.
The only leave that "may not be disapproved by a commander," is "maternity convalescent leave" (6. a. (7)).
The Directive doesn't state a commander may not disapprove the other leaves covered in the Directive.
The Directive "limits" secondary caregiver (SC) leave to 21 days, but doesn't mandate commanders grant 21 days non-chargeable leave.
This, and the "may approve" are the sticking points.
What to do:
You should consider using the Open Door policy to have a discussion with your COC. Determine what their position/interpretation is. You can ask the commander how they interpret it. If not satisfied, raise the issue with the IG.
The Directive states it "must be taken in only one increment," so you only get one shot at this.
Not sure what you mean with regard to the reference about use or lose leave being denied. This only muddies the water without further explanation.

COC can’t deny the 21 days of pat leave. See if you can find someone to cover your CQ. I would take it to BC or CSM before IG.

Sounds to me like maybe your COC isn't tracking the new change in Army Directive 2019-05. However, you need to understand it is NOT a requirement. Depending on mission requirements and manning you may or may not get the entire 21 days.