At about 22′58″ into the video linked below, the Red Sox's batter (Christian) swung and missed when the Yankees' catcher didn't get the ball either. If Christian didn't swing, could he get a walk? The link starts at 22′53″.
From the 2019 MLB rulebook a BALL is:
...a pitch which does not enter the strike zone in flight and is not struck at by the batter. If the pitch touches the ground and bounces through the strike zone it is a "ball."
In the linked pitch, neither of these are true (by my estimation). Whether the catcher catches the ball directly has no bearing on the calling the pitch as a strike or a ball. Assuming the umpire judges the pitch as entering the strike zone, this is still a strike even without the batter swinging.
There is a scenario on a dropped third strike that the batter is allowed to attempt to reach first base (rule 5.05(a)(2)), but it doesn't change how the pitch is called.