Question

Which battle led to the American recapture of the Philippines?

Answer 1

I would go for the Aero-Naval battle of Leyte Gulf.

Explanation:

This was a very big engagement where you had the entire might of the two fleets involved (one to defend the other to destroy the invasion forces):

If I remember correctly the Imperial Japanese Navy devised a plan where there was a bait: a fleet of carriers (Admiral Ozawa´s, basically without aircraft for lack of pilots and hardware anyway) luring the American fleet away from the invasion cargoes, assault ships and light carriers to let them vulnerable to the combined attack of 3 Japanese fleets (Admirals, Nishimura, Shima and Kurita) passing through the various islands and straits of the Philippine's archipelago and attacking, by surprise, the landing areas. Also airplanes from various land bases should have attacked in waves the American fleet.
The Japanese plan worked...almost!

The American fleet discovered the fleet of carriers and went north (admiral Halsey) to attack them leaving (also because a misunderstood message letting the fast heavy cruisers of admiral Lee to join the pursue) the landing zone vulnerable.

At this point, when the big guns where far away to hunt down the Japanese Carriers, Kurita arrived with his heavy cruisers and battleship at Leyte (out of Samar Island) and attacked the fleet of destroyers and light carriers (admiral Spargue): at this point amid confusion and panic the destroyers (the wolves!) turned against the Japanese fleet and attacked to protect the carriers! Then the escort destroyers (the cubs!) joined in in the desperate struggle! While they were dismantled by the fire of the Japanese heavy cruisers they gave time for the carriers to deploy smoke curtains and send in the few aircrafts intended as fighter-bombers to support the landings!

At the point when the decimated light fleet was on a platter to be annihilated and the landing zones free to be attacked...Kurita decided to withdraw!!!
No one exactly know why, but this decision saved the invasion troops and gave to the American the Strategic victory and the free hand to invade the Philippines.

[The wolves attack!]

By the way this battle presents the first appearance of organized Kamikaze attacks!