Of course he will. Not only will he call for the public execution of the British spy, he will probably insist that he is going to do it himself. After all, it's not like Trump is a rational man who thinks about what he is saying before he says it.
This is a silly debate question, but I don't think that Trump would bother with this guy. Trump's ego is certainly big enough to take the hit. If I were the British spy, though, I would stay on my own side of the Atlantic for the next few years or so.
President-elect Trump will not call for the public execution of Christopher Steele for releasing a dossier on Trump allegations. However, Trump may very well call for the prosecution of former British spy officer. Trump has long threatened to sue journalists and those that smear him with baseless allegations that hurt his character.
Trump will most likely accuse this spy of all manner of things but I doubt he would call for the man's execution. Britain is an ally of the United States and it would be extremely inflammatory if Trump wanted one of their citizens killed. I have to believe Trump will listen to his advisors on this issue.