Williams was shown the red card for a hard foul on Portland's Andy Polo in the first half of Timbers' 3-0 win against the Galaxy. Polo was helped off the field and will miss the season after rupturing his quadriceps muscle and tearing the meniscus in his left knee.
"The LA Galaxy are aware of hateful, threatening and racially abusive social media messages sent to defender Derrick Williams. This abhorrent language and behavior have no place in our sport or in our society," the Galaxy said on Friday.
"The club is working with Major League Soccer and the proper authorities to identify any and all abusers. We will take the appropriate next steps to protect Derrick and all of our players from these acts of hate."
Williams was born in Germany to an Irish mother and a Black American father, and he plays internationally for Ireland, where he grew up. The Galaxy signed the former Aston Villa prospect last March from Blackburn Rovers, and he has started four matches in his first season stateside.
The league has not ruled on whether Williams will miss additional matches for the foul.
Galaxy head coach Greg Vanney told reporters on Thursday that Williams was remorseful for injuring Polo.
"He reached out to [Polo] in the moment. He reached out to the player after. The loser right now in this is the player who is out for the rest of the year. Derrick is now suffering from some other things that should not be happening and taking place, whether it's social media or whatnot, that are complete nonsense as well.
"He's a good guy who, at the end of the day, got a tackle very wrong, and he knows that."