I spot-checked a few museum web sites for Madrid and Barcelona, and this is what I found:
National Museum of Catalan Art (highly recommended): Free for "Titleholders of accredited Cards for Disabled from the Generalitat de Catalunya".
Prado Museum: Free for "Visitors with physical impairments", but I didn't find any further explanation.
Reina Sofia Museum: Free for "Persons with disability. The person accompanying the disabled person can also enter the museum free of charge, provided that his or her presence is necessary during the disabled person's visit (with disability card issued by government agencies)."
Picasso Museum: Free for "Holders of the Metropolitan Pass for accompanying disabled people" and for "Holders of the free-of-charge Pink Card", but I don't know what the Pink Card is; it may have nothing whatever to do with disability.
So it does appear that there are some opportunities for free admission, but I'd expect to play it by ear and not be shocked if told that a document of some sort from a Spanish government agency was necessary. That requirement seems to apply to the NMAC and the Picasso Museum in Barcelona.
I found that admission people at Spanish tourist sites seemed to want to allow discounts. I was often asked whether I was 65 (not quite yet), and I got the feeling they were sort of surprised when I didn't simply say "Yes" and pay the lower admission price. So I think they'll do their best. But if a specific local card is required, I don't know what they'll do. It certainly won't hurt to travel with US documentation if it's available.
Also be aware that there is often a free-entry period at least once a week. The Picasso is a zoo at just about all times during peak season, so I don't think I'd want to try it when it's free. But the other places I listed are large, and I visited two of them on free days without problematic crowds.
I haven't been to Montserrat and don't know what sort of special challenges might be experienced by someone in a wheelchair. It's an easy sidetrip to do on your own if without special physical challenges, according to all the guidebooks.
If Enric doesn't show up here to provide input within a day or so, post a separate question with a title like "Montserrat in a Wheel Chair?". I'm sure that will get his attention.