What happens when I ask Google to go through everyone's gmail inbox and remove my information and all emails I've sent?
In your example you were the one that sent your information to some other 3rd party, so you would be the one responsible for that data transfer and its consequences.
(1) Companies that collect data and "process" it. If you're hosting it with reason, it's no big deal. If you're actively doing something with it, then you can run afoul of the law.
(2) Companies that share their information with third parties. It puts a much larger onus on companies that have your data to use it appropriately and only for intended purposes.
If you read through the wording of the law, it's perfectly possible to have an e-mail service that complies with everything.
I think the crux of this particular argument though is whether or not you "own" the emails you sent. I think at the most if you pushed this issue to the max you could get a company to scramble your email address so that it doesn't identify you anymore... but all of this is more a thought exercise about minute details. The true intent of the law is the major points above.