Not so much.
In the bigger picture the problem was laid directly on George McGovern who first failed to vet his VP, then claimed to be "1000%" in support and finally dumped the hapless fellow.
McGovern was severely punished at the polls by voters.
McGovern's failure to properly vet Eagleton and his subsequent handling of the controversy gave occasion for the Republican campaign to raise serious questions about his judgment.
In the general election, the Democratic ticket won only Massachusetts and the District of Columbia."
That was my memory - ECT is still kind of a shocking thing to have done (no pun intended). And look at the sentence after the one you bolded. It was the central part of the story.
Though McGovern didn’t lose because of it, he had no chance anyway. It was the last time (even) Minnesota went Republican.
I remember those days pretty well having been both a lefty and a McGovern voter at the time, even passing out flyers for him. (What can I say? I was young and stupid.)