Puigdemont has yet to declare full independence from Spain.
Speaking during a visit to Aberdeen, Ms Sturgeon said: "I think dialogue now has to replace the kind of confrontation we have seen".
The central government in Madrid said the referendum was illegal.
The United Nations rights chief said he was disturbed by the violence during a police crackdown on the banned independence referendum in the Spanish region of Catalonia and urged Madrid to investigate any possible misconduct.
Images of police using rubber bullets and batons on voters were broadcast worldwide over the weekend - prompting condemnation internationally and intensifying the anger that brought Catalans to the streets. "I will not hide, we are really anxious about Spain", he said. Most of the injuries were not serious and no officers were hospitalized.
He called for Spanish national police reinforcements to leave the region after Sunday's violence, and for worldwide mediation to solve the political deadlock.
Spain's Interior Minister Juan Ignacio Zoido said Madrid would "take all necessary measures" to stop the "intolerable harassment" of national security forces. Experts say European Union officials are fearful that supporting it could open the door for many other independence campaigns across the continent.
The region's pro-independence president, Carles Puigdemont, who has said an independence declaration will come in a few days, is due to deliver a speech later Wednesday.
Despite the police action, the Catalan government reported a turnout of over 40 per cent, 90 per cent of whom voted for independence. Forty-two percent of voters reportedly took part. "For many years during the Franco era, people weren't allowed to vote", Pique said. In all, around 850 people were sent to hospital, including 33 police officers.
Young people making noise by banging pots during a general strike in Catalonia called by Catalan unions in Barcelona, on October 3, 2017. The region has its own language and generates a fifth of Spain's 1.1 trillion euro economy.
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