There could have been. Henry VIII was not his family’s eldest son - his elder brother, Prince Arthur, died in 1502. Had he lived, there would have been a King Arthur.

In that specific case, the naming was deliberate; Henry VII had been crowned as a result of some extremely messy civil war business to end the Wars of the Roses. Having geanologists research his family line as far back as it would go, trying to identify the site of Camelot and naming his firstborn son (who was viewed at the time as a living symbol of a new peaceful golden age) after a legendary king after deliberately sending his wife to the site of “Camelot” (Henry identified Winchester, and sent his pregnant wife to Winchester Cathedral) so the boy could be born there...this was all part of a conscious attempt to strengthen the claim of the new House of Tudor.

But Arthur died before he could be crowned, so it didn’t happen.

Personally, I suspect there has never been another attempt to name a royal child Arthur for much the same reason as there has never been another Alfred. Whether Arthur the legend or Alfred the Great, giving a child such a name sends a deliberate message about the kind of ruler that child is hoped to become. It would be an absolute public relations nightmare if the child with the legend’s name grew up to be a walking disaster!

First can I say "King Arthur" is entirely fictional, no such person ever existed and you can forget any mumbo jumbo about a Roman general called Artorius as well. The tale was first started by a Welsh monk called Nennius who took a pagan god legend and wrote the deeds as if they had happened in history. Geoffrey of Monmouth used this as a source to create his great work of fiction which was probably the greatest historical hoax of all time, certainly it was the longest lasting. He created the actual story we know today. The caveat to all this is that the medieval royalty believed this story as fact so your question still stands I just wanted to clarify as it gives the impression you believe it yourself. It also explains why we can't have any more Arthurs pre 12th century, the story just plain doesn't exist before.

Second, Arthur is not an English King, he was Celtic or as the Medieval man would see it, Welsh. It was a bad thing to be Welsh in the mid twelfth century and it's likely, being a Welsh man himself, Geoffrey wrote the tale to create a celtic hero greater than English or French heroes as a result of prejudices he'd endured studying in England.

So the tale is only created in the 12th century when English aristocracy is at the very beginnings of forming it's own identity. At this time it still see's itself as Norman French not English and there is quite a shock when Henry III names his son Edward (an English not French name). There is also a strong convention to reuse family names, at this point, Edward I naming his 1st born son John after his grandfather and his second born Alphonso after his wife's father (yes England nearly had a king Alphonso!)

This sense of English identity among the aristocracy is there or thereabouts in the reign of Edward III who also is big on his Arthurian tales. If anyone was going to name their heir Arthur it would be him, but he too goes for family names and the lack of any English aristocrat naming their child Arthur does indicate the name was a bit unfashionable like names like Gurtrude are seen today.

Henry Tudor does name his first son Arthur. The Tudors are a Welsh dynasty and this is a deliberate ploy to use the Arthurian legend of a King coming from the Welsh hills to rule all of Britain. The lad dies and we are quickly onto the Scots in the 16/17th Century where the tales of Arthur are becoming unfashionable.