The whole point of a holiday request is to ask for approval or denial. An employee cant just TELL you when they are going to take holiday.
He knew the rules as well as the next Employee and I thinki most Companies work with a similar procedure.

He is part of a team, and by acting like a spoilt brat he is letting the rest of the team down. If he takes the time off and walks I think your team would be better off without him. So many people think they are indispensible

We have very strict rules at our place on holiday, mainly because if everyone didnt stick to the rules it would put a lot of unfair pressure on our collegues. All about give and take.

I think you're being more than fair offering to swap your week with his, you are actually giving him the option to take the holiday he so desperately wants.... he cant have everything his way, im sure if the shoe was on the other foot he sounds like he'd be one of the first to complain that its one rule for some and different for others.