Erik's answer is correct, however an example was requested in the comments. So, assume a character with dex 16 (+3) is wearing a breastplate (AC 14, max dex +2). The character's AC is 16 (14 from armor + 2 from dex)
On the first hit, the ooze damages the armor to AC 13 (+2 dex). Character AC is 15.
On the second hit, the ooze damages the armor to AC 12 (+2 dex). Character AC is 14.
On the third hit, the ooze damages the armor to AC 11 (+2 dex). Character AC is 13.
On the fourth hit, the ooze damages the armor to AC 10, which destroys it. The character is no longer wearing any armor, and the character now has an AC of 13 from their dex bonus alone.
In short, the armor's maximum dexterity bonus is entirely irrelevant to how long it holds up against a gray ooze. This also becomes clear when the opposite situation is considered, where the armor is light or medium armor but the character has a large dexterity penalty instead. Assume the character is wearing a chain shirt but has a dexterity of 4 (-3 modifier). The character's AC is 10, but the armor will still take 3 hits to reduce its AC from 13 to 10 and be destroyed.