Ice bath exposure does not increase cortisol levels, and if anything, it reduces them.

The finding

What the evidence shows

Part of this holds up. Some studies of whole body cold water immersion find cortisol unchanged or lower, including one that measured a drop about three hours after a 15 minute immersion at 10 degrees Celsius. But flatly saying cold does not raise cortisol is too strong, because the cold pressor test, a hand or foot held in ice water, is a standard laboratory stressor used precisely because it reliably raises cortisol. One study measured cortisol in saliva rising from a median of 0.14 to 0.34 micrograms per deciliter ten minutes after the task. The honest summary is that the answer depends on the protocol and on when you measure, not that cortisol never goes up.