Following the death of the Queen on Thursday 8 September and the proclamation of King Charles III there have been arrests around the UK of those who have appeared at official mourning ceremonies to voice their republican views about the Royal Family.
Walker - who recently left BBC Breakfast to host Channel 5 5pm news programme - tweeted: "I believe in the right to peacefully grieve. I believe in the right to peacefully protest. I fear there may a few uncomfortable meetings of the two in the coming days."
Walker, 45, met Queen Elizabeth II in 2012 when she opened the new BBC Sport studios at Salford in Manchester.
He recalled of the encounter: "It was a pleasure to meet The Queen... She was great fun and we had a lovely chat for a few minutes. Rare combination of making my mum proud and impressing the kids."
Following the announcement of her death he tweeted: "I don’t know about you but it feels very strange to be reflecting on the death of Her Majesty the Queen. She was so much a part our identity for so many generations. Life feels very different without her."
Walker left the BBC after 13 years after signing a deal with Channel 5.
The father-of-three said: "I've never ever been motivated by money in any job that I've ever taken, and even though I've loved working at BBC Breakfast and everybody who works on the show, this is a fantastic opportunity to work with a brilliant team at Channel 5."
He had hosted BBC Breakfast since 2016 and has been replaced by Jon Kay.
Walker competed in the 19th series of Strictly last year and finally left the BBC dance competition with professional partner Nadiya Bychkova in the quarter-final after weeks of evading the dance-off thanks to votes from viewers.
Before he joined the BBC Breakfast team Walker presented Football Focus on BBC from 2009 to 2021.
Watch: Dan Walker's debut on Channel 5 news