I did a huge research to see if there is a way to find out. The answer is probably "No", but, you may be able to get some positive results. Here are my findings:
From within Joomla, it is not possible to do do that.
From within the server, it is somehow possible to do it by examining the Apache access logs, but with great difficulty.
You will need to examine the logs and do a grep for registration.activate, which means that this activation was done by the user. Now, the issue is, you won't be able to find which user it is because the hash value is erased from the database once the user is activated. However, there might be some "signs" that will tell you, if you examine the logs very closely, which user is that (maybe track the IP and check its activities).
Of course, doing this manually is only practical if you have a few users every week. Otherwise, you will need to develop a script for log analysis (look at awk).
Your question is by far one of the most interesting I have ever attempted to answer, and I literally researched it for a couple of hours.