Turning a permanent face-up by paying its morph cost is a special action that can be performed anytime a player has priority and doesn't use the stack. It can't be countered, nor is it an activated ability.
Follow up question: Can the morph cost be countered? Or is that an ability?
Neither, it's a special action that doesn't use the stack. Since it doesn't use the stack, there's no way to stop it after the controller declares the action.
702.35d If you have priority, you may turn a face-down permanent you control face up. This is a special action; it doesn’t use the stack (see rule 115). To do this, show all players what the permanent’s morph cost would be if it were face up, pay that cost, then turn the permanent face up. (If the permanent wouldn’t have a morph cost if it were face up, it can’t be turned face up this way.) The morph effect on it ends, and it regains its normal characteristics. Any abilities relating to the permanent entering the battlefield don’t trigger when it’s turned face up and don’t have any effect, because the permanent has already entered the battlefield.
If you suspect your opponent will flip a morphed creature, you're welcome to try and prevent them from being able to pay the cost (E.G. Destroying all their creatures if the cost is sacrificing a creature), but you can't counter it.
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Once a card has been turned up (or if it enters the battlefield without the morph being done), can the original cost (on most of my morph cards, it's three colorless), be used to turn them face down, or does it have to be cast when it enters the battlefield in the first place?