West Brom confirmed that Livermore reacted over remarks made about his son, who tragically died shortly after birth in 2014.
The England international waded into the crowd after hearing the comment before the incident was broken up by stewards and coaching staff.
Livermore was eventually escorted down the tunnel by a member of the West Brom backroom staff, as assistant head coach John Carver remonstrated with stewards.
West Ham identified and ejected the supporter at the time of the incident and are still investigating exactly what happened.
The Hammers are already committed to banning the individual from the London Stadium and are now just considering how long that will be imposed for. A lifetime ban has not been discounted.
Livermore has decided not to refer the incident to the police, with West Ham promising to take the strongest appropriate action.
A West Ham spokesman said: "West Ham United can confirm that the individual involved in an altercation with Jake Livermore during last night's match against West Bromwich Albion was immediately identified and removed from London Stadium.
"After the game concluded, West Ham United officials liaised with West Bromwich Albion, Jake Livermore himself and the Metropolitan Police to commence a full investigation into the incident.
"The player has today confirmed he does not wish to make this a police matter, however, given the nature of the allegations, the club will continue to investigate before taking the appropriate action."
In a statement released on their website, West Brom said: "In the light of intense media speculation, West Bromwich Albion would like to clarify an incident involving our player Jake Livermore in last night's Premier League game at West Ham.
"Following his substitution in the second half, Jake was subjected to general abuse from home supporters which he shrugged off. However, he acknowledges that he moved to confront one spectator who chose to make a comment about the death of his infant son.
"The club would like to think that all right-minded football supporters could understand this reaction which did not result in any physical exchanges before Jake was led away to our dressing room.
"Jake is an outstanding young individual who has the total support of everyone at the club and he has made it clear he considers this statement to be an end to an unhappy incident.
"Albion were very appreciative of the swift response by the officials of West Ham in dealing with this matter and are more than happy to leave any further action they deem appropriate in their hands.
"The club has responded to an FA request to offer our observations of the incident and will not be making any further comment."
In Scotland, the distasteful missiles were aimed at Dunfermline midfielder Dean Shiels during Tuesday's second-tier Championship game. An accident led to Shiels undergoing surgery in 2006 and losing sight in his right eye.
Falkirk says it "condemns the abuse witnessed towards Dean Shiels in the strongest terms and will be working closely with Dunfermline Athletic FC to identify those responsible."
In a statement, Falkirk adds that it "will look to take action against anyone identified ... and will not allow this disgraceful behavior to be repeated."
A previous game between the teams in October was marred by two Falkirk players — Joe McKee and Kevin O'Hara — taunting Shiels. O'Hara was banned for eight game and McKee for four games.
