NATO does “authorize” military operations - the NATO ambassadors meet and agree to take military action, which means that NATO assets and staff can be used to help plan and support the action. NATO never authorised the invasion of Iraq, so NATO’s own assets and facilities could not be used. By contrast, Libya and Kosovo were NATO authorised operations, so NATO resources could be used.
NATO’s own resources are essentially office buildings, computer and communication systems, bureaucrats, and seconded military planners. The US didn’t need them for the Iraq War, although it would have liked to have had them involved, because of what that would have meant politically/diplomatically/symbolically.
Despite claiming a broad “coalition”, only three countries actually contributed fighting assets to the initial US invasion - the UK, Poland, and Australia - only the first two are NATO members, and Poland at the time had only recently joined (only 4 years prior). 3 more NATO members (Netherlands, Italy and Spain) provided non-combatant support. I don’t see in what meaningful sense it could be said to be something NATO did.
More NATO members joined in the occupation forces after the initial invasion. But that was never an official NATO operation because those require unanimous approval, and certain member states (most notably France) were always opposed to that.
NATO eventually did become involved in Iraq - training the new Iraqi government’s military. But no fighting ever happened in NATO’s name.