There may have been a misunderstanding between Apple support and you, but I assure you they will not/cannot DENY service. They were only denying you a replacement not repair. If you called them right now and said you wanted to have your computer REPAIRED because Amazon will not offer you an exchange, they would happily guide you through the repair setup process.
Williape is certainly right... either a misunderstanding, or an error on the part of AppleCare. Your computer came with a warranty from Apple. You should have a copy of it included with your computer.
Unless it was damaged during shipping or something (btw, what IS the problem with your computer anyway?), then Apple will take care of it for you.
Do you have an Apple Store nearby?
Just bring a copy of your receipt with you, and the computer to the store (make an appointment first).
BTW, don't even MENTION Amazon unless Apple asks where you bought the computer, because Apple really shouldn't give a rat's ass about where you purchased it from, as long as it's an authorized retailer, which Amazon is.
Let us know what happens.