It doesn't matter whether or not the OP needs to personally apply for a license. The fact is that you cannot be a resident without some sort of medical license - temporary, state issued or institutional. If your program applies for you (as mine did) using the documentation they have, then you may not have to send in your diploma before July 1 but they WILL want it at some point in time. My program applied for me but I had to submit the diploma as soon as I had it so that the license could be finalized (we had training and regular licenses in PA).
This will vary from state to state and program to program. Do you need it before you start residency? I would venture in most cases yes because programs are supposed to verify that you are legitimately able to start working and many don't want any unpleasant suprises on July 1. Medical students have been known to lie and schools have been known to not provide the documentation needed to start residency in a timely fashion.
If your program waited until after you started residency to request a copy of your medical diploma its because they knew they needed it to obtain the license. My programs would not have allowed someone to start without a copy of the diploma...programs have been burned before. I think its probably more useful to tell the OP that there are situations in which you have to present your diploma before you start residency rather than have him assume its *probably* ok.
This is indeed the case. I think there might be a few states that do not require a training/educational license, but most do and those that do not are being pressured to make this a requirement.
I think you need to be clear on one thing. You do need to graduate before you start work, even if you are in one of the fortunate few states that do not require a license if they even exist anymore. There are severe consequences for beginning a residency without the appropriate credentials in hand. (In one state, denial of permanent licensure, fines and the threat of jail time).
In recent years there has been "orientation creep" where orientations have been getting earlier and earlier. And some evil institutions have started working people the minute the mid-June 2 day orientation is over. So you must be very sure you have the appropriate credentials to start a residency on the day you actually do start.
I am guessing that your concern is can you participate in the match as a US senior in the year you graduate even if you graduate on June 26. The answer is likely yes. However, if you need to stay at your school and complete some aspect of your education, you may find yourself compromised on the orientation issue. This is why compression of the time between medical school completion and residency commencement is a bad idea. People do need time to move, find a place to live, and get settled in their new digs before they become sleep deprived zombies.
For programs like mine, where a TY year is required, I had to finish my first year on June 28, start my residency on July first and drive 1740 miles between the two.
If my residency had any special orientation I'd have been sunk. My PGY1 program knew I'd be overseas doing humanitarian work until shortly before I started, I discussed it with the program director and they were very accommodating of the vagaries of international travel from countries that we were not on such good post-9/11 terms with. If your selected/matched program is a good one, it will work with you given your circumstances.
If it won't then perhaps you may wish to reconsider whether you really want to be at a program like that and re-do your rank order list.
Another thought: if there is any doubt at all that you will not complete medical school on time to begin residency, meaning you may not complete until July 1, then perhaps you should discuss this with your dean and not reset your schedule to complete in July, and enter the match a year later. The down side of this is that you will take 5 years to complete medical school and I vaguely remember that there was a question back in the dark ages on one of the residency apps I filled out about extended medical school time. And, of course giving up a year of your life, in waiting, which is no small price, in my mind.
I think if you are assured of a graduation before July 1 (ie June 30), you should be fine. But this is an opinion only.