Here's a nice discussion with some literature references. My take-away is that even within SPS the effects are species-specific, but most focus seems to be on skeleton formation rather than tissue damage per se.
Corals Live Corals Aquarium Saltwater
www.queencitycorals.com
In particular, one of the papers referenced seems to indicate that pH changes in this range don't seem to affect the coral tissue metabolism (photosynthesis rates etc) very much, and the authors note that other studies have been equivocal. But the calcification rate in the skeleton is more strongly affected (with acropora seemingly more sensitive than a stylophora, pocillopora).
The resilience of corals to ocean acidification has been proposed to rely on regulation of extracellular calcifying medium pH (pHECM), but few studies have compared the capacity of coral species to control this parameter at elevated pCO2. Furthermore, exposure to light and darkness influences...
www.nature.com
But its an intriguing idea - maybe there's some other injury/immune compromise not measured here that makes them vulnerable to infections etc. Most likely one of many factors.
Interested to see the perspectives of others here.