No. We used to do that, but we found that most of the time when somebody uninstalled, they were planning an immediate reinstall. At which point they'd discover they had lost the email with their registration key. So we got lots of support emails requesting a replacement key. Lots and lots and lots of emails -- often 100+ per week. Since changing it to require resets on our end, the emails have dropped to < 10 per week.
(On a related note, we used to have an "Unregister" button in the dialog. It turned out that like a "wet paint" sign, users were unable to resist clicking it to see what would happen. And discover they had lost their registration email. Request replacement registration key. Click button again to see if the first time was a fluke ...)