Earlier Wednesday, the Times-Picayune quoted an unnamed player saying that Spagnuolo should be fired.
Vilma called the report "'anonymous source' bs" on Twitter and then refused to address specifics of the player's comments when contacted by the Times-Picayune.
"I'm bothered you reported it. We're not the Jets who run to the media for everything," Vilma told the newspaper.
The Jets were plagued all season by anonymous players sounding off in the media, particularly about backup quarterback Tim Tebow.
Vilma wouldn't give his take on the comments but said the player should have attached his name to the statements if he believed them.
"That's not the question or the point. If he's man enough to tell you, he should be man enough to put his name on it. And you should do the same," he told the newspaper.
Under Spagnuolo, New Orleans' defense was historically bad in 2012, allowing the most yards (7,042) ever in a single season. The Saints also finished last in total defense, rush defense and 31st in pass defense.
"To give up what we gave up can't be all talent," the unnamed player told the newspaper. "Look at where his units (have) been ranked before. I think one top 10?"
The player said Saints defenders didn't have a say in Spagnuolo's defense.
"He does have that good-guy persona, but he is a control freak and treats people like crap," the player told the newspaper. "(Spagnuolo has) no patience and zero personality. (He) has a way of pissing players and our defensive coaches off with how he says and does things. (I) think it's even harder after having (former defensive coordinator) Gregg (Williams), who guys enjoyed."
Linebacker Curtis Lofton, who like Spagnuolo just finished his first season with the Saints, told the newspaper that he believes in his defensive coordinator's system and called the anonymous player "a coward" for not attaching his name to his remarks.
"It was (Spagnuolo's) first year putting it in, and everything we went through, yeah I'm a firm believer in the system," Lofton told the newspaper. "I think it's a complicated system, but at the same time, it has an answer for everything an offense does. We will be successful in the future. ... Yes, there is some thinking that goes into it before the play and during the play, but once you've got a grasp to it, I really think this is a great system."