Quite an oration, rhapsomatrics! I hope you don't mind, but I took the liberty of making some minor, but necessary, modifications.
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WELCOME Tracy!
Personally, if I had seen the title "The Americas Manager" (without the apostrophe) it wouldn't have bothered me at all. However, "The America's Manager" implies only one of the American continents

(so this one is not an option). America is singular, Americas is the plural form. To use an apostrophe to show possession for the plural form that ends in "s" you merely put it after the final "s". The point of possession is sticky here, but when I look at "The Americas' Manager" I think of "The Manager of The Americas" (so this one works). "The Manager of The Americas" sounds the best, even though it is longer, and it evokes less confusion, in my opinion.
Let's wait to see what the other grammarians say. . .
Smiles!

SMS