That would certainly an interesting audio clip to listen to! The actual site here (liveatc.net) keeps regular archives of only recent audio within the past 45 days of live feeds. So, if this United flight was more than 45 days ago, or was in an area which this site wasn't set up for then it doesn't exist in the archives.
On the other hand, many users here find and post clips in the "Audio Clips" section... if it was recorded somewhere, there's a chance it's in there. Happy hunting!