No API - I worked on one last year, but twas never released. Hoping to see something within the year - not just for images, but also collection data (object records, for instance).

Many other institutions have open APIs for their collection data, and often for images as well. See:

Cooper-Hewitt (data released on github) Brooklyn Museum (API) Powerhouse Museum (API) Rijksmuseum Indianapolis Museum of Art (also data on github) Europeana (a EU-wide consortium) DPLA (similar to Europeana, but in the US)

Also, lots of other institutions have open access to their images - many much more comprehensive than The Getty (not to sound bitter, but Getty is getting a lot of press for this where others have lead for years). It's really an imperative at this point - and for most institutions more a matter of when (since few have substantial technical staff) than if. The Getty is an exception, having a large, well-funded staff, but tends to be conservative in policy changes.

For other examples of cultural institutions sharing images widely:

National Gallery (high rez) Los Angeles County Museum of Art The Met

All this just off the top of my head -