The Sun led the news of Hodgson’s appointment as England manager with the headline "Bwing on the Euwos! (We’ll see you in Ukwaine against Fwance)", in clear reference to the West Brom manager’s speaking voice.
The headline led to heavy criticism of The Sun on social networking sites, with many fans upset at both The Sun’s immediate antagonistic tone and the poking of fun at speech impediments.
But, while a "large number of objections" had been received both by the FA and the PCC, English football’s governing body would not be making an official complaint about the matter.
FA Chairman David Bernstein said: "We are delighted at the media response to Roy’s appointment but are disappointed with the headline in The Sun, which we consider is in poor taste and disrespectful."
The FA’s statement added that it had "raised it with the newspaper and made it clear that their front page is unacceptable to us."
Hodgson signed a four-year contract on Monday, succeeding Fabio Capello, who resigned after disagreeing with the FA over the removal of John Terry as England captain.
The former Fulham, Liverpool, Internazionale and Switzerland boss has managed many club and national teams in a 36-year career, but the tabloid media - led by The Sun - had been campaigning for Tottenham's Harry Redknapp to be England's next manager.
Hodgson will remain West Brom manager until the end of the season.