Around 100,000 demonstrators took to the streets demanding a ceasefire in the Israel-Hamas war from noon on Saturday.
Graffiti was daubed on the Field Marshal Earl Haig Memorial, on Whitehall, using red paint.
According to Historic England, it is a Grade II-listed statue.
Haig, who died in 1928, was commander of the British forces at the Somme and Ypres during the First World War, and his role has made him one of the most controversial military figures of the last century.
He eventually won a battle of attrition against Germany on the Western Front.
But he became a much-maligned historical figure for his controversial strategies which led to excessive casualties of British troops under his command.
He founded the Royal British Legion Scotland and the Earl Haig Fund – now known as Poppyscotland – in 1921.
