I thought I could save money by buying a car part at an auto store and having a garage install it. But they refused because I didn't buy the part through them ("You don't bring your own food to a restaurant and ask them to cook it."). Are they correct?

Yes, they are correct. Restaurants DO supply their own food.

Seriously, it is probably a reliability issue: They know their supplier is trustworthy, they don't know yours is. Not to mention, they get to charge extra when they buy it themselves, and they don't want you to be able to take that from them...

I understand that if the repair turns out badly then the garage, the parts store, and the part manufacturer could all be involved. If you let the garage put a part on your car that they provide, it's a one-stop deal and they will most likely stand behind the part and the labor.

Like it's been mentioned, they most likely get to make more money off you if you bought the part directly from them. It's happened before and I just told them (Firestone) I can find someone else to install it easily and they quickly changed their mind.

I remember one repair center said what my problem was, but the owner said "We're out of the part, hit this supplier and we'll install it for $25" (Midwest) so I did.

Some places are very tight, others are a bit nicer. :)

If they sell it themselves, yes. If not, then they, along with any other business, have the right to refuse you service for any reason they feel, except for discrimination because of race, religion, etc.

If you think about it, if you have a business and certain suppliers, and you begin to accept business from people who have supplied their own whatever, wouldn't there be a very good chance that your supplier will not give you the discounts you can pass along to your clients? (And, yes... If they get discounts, most WILL pass them on. If not, many only charge the "trip fee" that they are charged to get the item, if they are charged one.) There may also be a reliability issue.

Most garages make their money from the labor, not the products they sell. And it's NOT really that much, considering the overhead and insurance required.

That's not surprising. There is one thing you can do instead. I bought a car part myself and got a mobile mechanic to fit it. He just came round to my place and worked in the carpark.