The family of Drummer Lee Rigby said the soldier would not want violent attacks carried out in his name, amid increased tensions between political and religious groups.
His relatives urged those who wanted to pay their respects to do so in a peaceful manner.
This weekend a number of marches are planned, including one by the British National Party (BNP) in the centre of London and another by Unite Against Fascism in Woolwich, south-east London.
BNP members had wanted to hold a rally in Woolwich, but police stepped in to prevent this, to avoid inflaming "current community tensions".
It comes as the second man accused of his murder, Michael Adebolajo, 28, was discharged from hospital and remanded in custody. He will appear in court on Monday.
Yesterday, Drummer Rigby's mother Lyn, stepfather Ian, wife Rebecca and son Jack, released a statement through the Ministry of Defence.
They said: "We would like to emphasise that Lee would not want people to use his name as an excuse to carry out attacks against others. We would not wish any other families to go through this harrowing experience and appeal to everyone to keep calm and show their respect in a peaceful manner."
The family said they were deeply touched after visiting Woolwich and thanked those who had given them time and space to pay their own tributes.
They added: "We all loved Lee deeply and we know that he loved us. We all miss him so much.
"Lee loved life and he loved people. He had many friends from different walks of life, some with different religious beliefs and cultures. But this made no difference to Lee. He always treated others with the greatest respect."
The family said the global support they received had been "truly staggering" and was giving them the strength to carry on.
An inquest has heard Drummer Rigby was returning to his barracks after a day working at the Tower of London at the time of his death.
Detective Chief Inspector Grant Mallon, from Scotland Yard's Counter-Terrorism Command, told the inquest, which was opened and adjourned on Thursday, that he was hit by a car before being attacked by two men armed with a cleaver and a knife. The court heard the 25-year-old was identified by a dental expert.
Meanwhile, a 25-year-old man has been charged by police in Inverness after an inflammatory remark was allegedly posted on a Facebook page commemorating Drummer Rigby. A message was removed following heated responses to a post.
A Police Scotland statement said: "Posting comments and threats may inflame the situation and could be considered as criminal in themselves."
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