Does making an ability check take an action?

The only ability checks that are listed as an action in the game are Dexterity (Stealth) when taking the Hide action, and Wisdom (Perception) or Intelligence (Investigation) for the Search action. Do any other ability checks require an action?

My fighter was chasing a floating target, and needed to climb up a short but steep cliff. The climb and horizontal run to the target were within his movement for the turn, but the DM had me roll a Strength (Athletics) check to complete the climb. I was successful, but after arriving at the top and attempting to grapple my target, the DM declared that I had already completed a skill check, which was an action. I was still able to grab the target using Action Surge, however I still felt slighted, as I thought the climb shouldn’t have taken an action.

Who was right here? what if the ability check was an Intelligence (Arcana) check to recall some information, should that still require an action as well?