I don't know if this will be enough so if it is not then don't blame me for it. Other official means are probably better or not ( i don't know).
Take a continuous video of yourself with Time.gov and show around where you are at the moment. Like If you live in LA, video yourself from your computer showing the time on the time.gov website ( and yourself)and you can go out and try to take a video of the street name maybe or your address, ... ( basically as a proof that you are in USA) . You get the point. But try to make sure it is a continuous video and not in parts.
Again i have no clue if this will do the job and probably other official documents are the way to go to show you are here but you can give the time.gov site a shot if you want.
that being said, again i have no idea if this is enough or acceptable form of evidence to show if you were here on June 15 or not. Also there is also another long thread discussing about how to prove you were here on June 15th in this link
http://dreamact.info/forum/showthread.php?t=27992 .