The seven-time world champion fought back from a 10-second penalty to clinch victory following a controversial collision with rival Max Verstappen, which left the Red Bull driver in hospital.
Facebook, which also owns Instagram, said it had removed a number of comments from its platform in relation to Hamilton, who has been a huge advocate for diversity and the Black Lives Matter movement, and called the abuse "unacceptable."
"In addition to our work to remove comments and accounts that repeatedly break our rules, there are safety features available, including Comment Filters and Message Controls, which can mean no one has to see this type of abuse," a Facebook company spokesperson told CNN Sport.
"No single thing will fix this challenge overnight but we're committed to the work to keep our community safe from abuse."
Meanwhile, Twitter said that racist behavior "has no place" on its platform and strongly condemned the abuse Hamilton received.
"We have clear abusive behaviour and hateful conduct policies in place and we take enforcement of these policies very seriously," a Twitter spokesperson told CNN Sport.
"We will take decisive and swift action on any account that breaks our rules."
Hamilton's team Mercedes, Formula 1 and the sports' governing body FIA also released a joint statement condemning the racist abuse "in the strongest possible terms."
"These people have no place in our sport and we urge that those responsible should be held accountable for their actions," it read.
"Formula 1, the FIA, the drivers and the teams are working to build a more diverse and inclusive sport, and such unacceptable instances of online abuse must be highlighted and eliminated."
It comes a week after three England players -- Marcus Rashford, Jadon Sancho and Bukayo Saka -- were subjected to racist abuse after missing penalties during the Euro 2020 final against Italy.
On Monday, Mercedes team principal Toto Wolff told BBC Radio 4 that the abuse of his driver was "not acceptable" and promised to "react to it."
"We have seen it in the football in the European Championship and the abuse that is absolutely not acceptable, and the same yesterday, and I still question whether some just don't get it," he added.
Red Bull also released a statement saying it was "disgusted and saddened" by the abuse Hamilton received on social media.
"There is never any excuse for it, there is certainly no place for it in our sport and those responsible should be held accountable," it read.
Around 140,000 people watched from a packed out Silverstone as Hamilton and Verstappen collided in a dramatic start to the race.
Hamilton, who was beaten into the first corner by his rival, made a number of attempts to overtake Verstappen on that first lap and when the drivers arrived at Silverstone's Copse Corner, the Briton's front-left wheel touched the Red Bull driver's back-right tire.
Verstappen spun off the track and across the gravel before crashing into the barriers. The Dutchman was taken to hospital but did not suffer any major injuries.
After the race, Verstappen took to social media to express his feelings about Hamilton's "dangerous" driving.
"First of all: I am glad I'm ok. It was quite an impact at 51G but feeling better," he wrote.
"Obviously very disappointed with being taken out like this. The penalty given does not help us in any way and doesn't do justice to the dangerous move Lewis made on track.
"Watching the celebrations after the race while still in hospital is disrespectful and unsportsmanlike behavior but we move on."
The comments were echoed by Red Bull team principal Christian Horner who was visibly furious when discussing the incident after the race.
"Lewis [Hamilton] has got more than enough experience to know that is unacceptable," he told Sky Sports.
"I'm just very disappointed that a driver of his caliber should make such a move like that. It's dangerous, it looked desperate."
Hamilton sent his best wishes to Verstappen on social media after the race but said he had raced "fairly."
"Today is a reminder of the dangers in this sport," he wrote on Sunday. "I send my best wishes to Max who is an incredible competitor. I'm glad to hear he is ok. I will always race hard but always fairly.
"My team showed grit and perseverance out there. It's a dream to win in front of my home crowd."
He was backed up by Wolff who described the collision as a racing incident.
"It always takes two to tango and these two are not giving each other an inch and it's a high-speed corner, and that's why these things are nasty to look at," he told Sky Sports.
Race stewards handed Hamilton a 10-second penalty for the incident but the Brit made his way back to the front in a dramatic conclusion, overtaking Charles Leclerc on Copse Corner with just two laps to go.
Hamilton's win gave him 25 points to put him just eight behind Verstappen in the drivers' standings as the rivalry between the two intensified.
All eyes will now be on Hungarian Grand Prix which will take place on August 1.