The 28-year-old England international charged into the stands shortly after his substitution in the second half of West Brom's 2-1 defeat by West Ham at the London stadium, before being restrained by stewards and escorted down the tunnel.
Livermore lost his newborn son Jake Junior in May 2014.
"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," West Brom said in a statement.
"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."
West Brom thanked West Ham officials for their quick reaction to the incident and confirmed that they had responded to a request for observations by the Football Association.
