I believe that James' cramping was primarily due to the heat of the building. Lebron James is one of the best players in basketball today. He doesn't have a history of injury like some of these other players. The building was simply too hot for him and it began taking a toll on his body.
The fact that heat forced Lebron James to miss the last seven minutes of the game shows that heat had an affect on the outcome. However it's important to remember that the heat was equal for both teams so its effect on play shouldn't be deemed an excuse for the Heat's loss. They simply got outplayed in Game 1.
LeBron James is a professional athlete who lives and plays in Miami. He should be well used to exerting himself for long periods of time in warm weather. Every other guy was able to handle the heat. Seems like he is a bit out of shape at the end of the season and looking for excuses.
Since both teams were equally exposed to the heat I don't believe that one team got an unfair advantage so I think it would be difficult to say that it effected the outcome of the game. While it is unfortunate that Lebron got heat cramps that could have easily happened to any player on either team. Injuries happen during sports all the time.
The heat didn't cost the Heat the game. If Lebron was so affected by it that he can blame the heat for giving him cramps, then why wasn't anyone else affected? It's just like earlier this year when Lebron said he couldn't play as well in the short-sleeve jerseys, or when he complained that he had to wear his custom made face-mask. If Lebron is that sensitive to his playing conditions, then maybe he should rethink his career...