So let me count out my objections to his claim:
1. Levee has been used in North America since the 18th century. (Orig. AmE) dyke has only been (slang (or a hyponym) for 'lesbian' since the 20th century. So, Americans definitely didn't start saying levee to avoid association with lesbians.
2. If you're an American like me, you primarily know levee from Don McLean's song American Pie, where it is a convenient rhyme for Chevy. (This is at the top of the 'Lynne's most hated songs' list. I hope I haven't earwormed you with it. My day is ruined.) I think of it as a Louisiana thing (it is used all the way up the Mississippi River), and that's where it came into English, from French. (That part of the continent came into the US with the 1803 Louisiana Purchase.) It may be more common now that people have heard it more in the news because of extreme weather in the Gulf States, but I still think of it as a vaguely regional term, rather than pan-American.
3. I only really knew the word dike from the story of the Little Dutch Boy. Where/when I grew up, I'd've called it a little dam (because we weren't put off by the homophony with damn either!) But note the spelling. The main American spelling of this thing is dike, whereas the 'lesbian' sense is usually spelled/spelt dyke, which which Merriam-Webster lists as 'chiefly British variant of dike'. So, in printed form, at least, the 'taboo' sense and the 'built up bank by the river'
4. It’s not like the British are freely going about saying dike for the meaning 'levee'. They tend to prefer the word embankment for such things. Who are the prudes now?
(The green (more BrE) bits above were added after first posting.)