No that does not make you a Spaniard, at least not in my opinion. Personally I don't get it when people say that they are Spanish, Irish, Italian or any other nationality when they have some ancestors that came from there, yet they themselves don't have that nationality, often they have never even been to the country, know very little about the country and don't speak a single word of the language.
I realize the phrase "Latin Americans aren't Spaniards!" could come off as defensive. It wasn't supposed to be. It's simply that you wouldn't call the Quebecois "Frenchmen", would you? French speakers, yes, but "Frenchmen"? No. And the Quebecois are at least descendents from French settlers by and large. Latin Americans have diverse roots (many European countries - not just Spain, native peoples, Africans) and so they are not Spaniards. Hispanic and Spaniard are not the same in English. Hispanic refers to people who speak Spanish or whose ancestors spoke Spanish, while Spaniard refers to a citizen of the country of Spain.