If you didn't get such a license, or such a transfer of copyright ownership, you are almost certainly violating his copyright.
It is completely false that creating something for payment automatically results in loss of the copyright to the client. Under US laws, you would need a properly written contract and only certain types of works even qualify as "works made for hire" when done by anyone other than your own employees.
The legal author of the work is the copyright owner. 17 USC § 201.
If you have any documentation digital or otherwise that proves he knew the only compensation he was receiving was a shoutout then no, he's entitled to nothing. If you can show the agreement was adequately suggested, you might win in court. My advise is to print your logo on paper, take it to the post office and mail it to yourself. This will at least create a copy of the logo that's been date stamped. This of course hinges on him not copyrighting the logo to begin with. Better yet, take it and copyright it now. The chances are he hasn't done that. if you can get it copyrighted before him, you can tell him to go pound salt.
If you had clearly mentioned in the terms and conditions page that the winner must hand over the ownership of the copyright and that he would no longer be entitled to use that logo and cannot claim any royalty, then you should not be worried about anything. In case, you’d not mentioned anything of that sort, as per the U.S laws you’ll have to pay him royalty.
If you clearly defined in the rules of contest that the winning entry became the sole property of your company, in exchange for the "shout-out". By entering the contest he waved all rights to ownership.
But. If there was any ambiguity in the wording of the rules and you didn't follow up by getting a sighed contract, you have a problem which could likely wind up in court with you losing.
..Intent to receive full ownership isn't enough to win a court case, by fulfilling the promised "shout-out" you already conceded the fact that he created the logo and was the de-facto owner at the time of submission.
If he created it exclusively for you, for a reward or reimbursement in kind then you own it, so long as you have given him the consideration you promised him, ie, the shoutout.
It was a pretty basic contract and no doubt a copy of your offer is available. You have completed the contract and he has no further claim.
This involves my company on Instagram. I held a contest that says whoever can make the best logo for me gets a shoutout. After a lot of submissions, I chose one and gave him the shoutout as promised.
However, now that I am expanding my business and have put the logo he made on t-shirts, he sent me a message saying he wants royatly for every shirt I sell since it's his logo.
My question is; do I have to buy him royalty? Or is the fact that it was for a contest cancel that out?