Could this exclusivity result in an anti-trust law suit?
Of course not. Apple and Amazon are competing. Apple just managed to get an exclusive deal to gain a competitive advantage (and yes, it is legal to beat your competitors. It's not legal to prevent them from competing). Amazon is free to offer an exclusive deal say to the Rolling Stones or to whatever band they choose to offer such a deal. "Music by the Beatles available from a download store" is not a market. Not even "Music by the Beatles" is a market, and if it was, then Apple and Amazon would still be competing in the market.