Police Scotland said three members of the public were injured during the disorder in Glasgow.
Three men – two aged 18 and one aged 31 – were arrested and charged in connection with various offences in the city during the evening.
Officers also attended a number of smaller, but peaceful demonstrations in Edinburgh, Falkirk, Perth, Ayr and Paisley, and no arrests were made.
ACC Waddell said Police Scotland were “prepared to deal with any escalation in disorder” over the coming days.
He added: “Officers responded to disorder and violence, including incidents in Glasgow where members of the public were attacked because of the colour of their skin. Officers were also attacked.
“I would strongly condemn that offending and send a clear message that there is no place for racism and violence in Scotland.
“We understand the concerns people have about their communities and will always balance the right to freedom of expression with the need to tackle crime without fear or favour.”
Widespread violence took place in Belfast on Tuesday evening.
Houses, cars and a bus were set on fire and all public transport was suspended in the city.
The Police Service of Northern Ireland (PSNI) said it had requested “mutual aid” from other forces across the UK.
Police Scotland said it would support arrangements, which could see officers from the force deployed if necessary.
In Glasgow, demonstrators with their faces covered chanted and carried Union flags and banners.
Officers from Police Scotland were deployed and diverted the crowd away from the city centre.
A number of streets and bridges were closed.
