The sport's body said in a statement on Monday that it had accepted an apology from Smith but "reiterated that this type of behaviour is not acceptable and Louis is fully aware of the consequences of any further indiscretions".
Smith, who retired in 2013 but returned to competition a year later, is hoping to compete at the Rio de Janeiro Olympics in August after taking a pommel horse silver medal at his home London Games in 2012.
The 26-year-old also won team bronze in 2012 and a pommel horse bronze in Beijing in 2008.
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However he lost out to team mate Max Whitlock, the 2012 pommel horse Olympic bronze medallist, at the British artistic gymnastics championships on April 10.
"I aired my frustration, verbally and through social media because I genuinely thought I had performed the best routine of the day," he said in a statement published on the British Gymnastics website on Monday.
"Having had time to cool off, and following conversations with British Gymnastics, who have helped me see the consequences of my actions, I realise the way that I expressed my feelings was wrong and it should never have happened."
Smith apologised to his team mates, the fans and judges for any offence caused and said he would get back in the gym to focus on the European championships and Olympics.
The individual members of the British team are due to be announced in July.
"Being a gymnast can be difficult because the sport is subjective; coupled with that I am very driven, and passionate about the sport that I love," said Smith.
"None of that is meant to be an excuse for my behaviour, but I am human and I let my emotions get the better of me."
(Reporting by Alan Baldwin; Editing by Alison Williams)
