Ms Park was forced from office on Friday, after judges unanimously upheld parliament's decision to impeach her over her role in a corruption scandal involving close friend, Choi Soon-sil.
Despite the ruling, she remains inside the presidential compound.
Thousands turned out for rallies in Seoul on Saturday, a day after three people involved in protests died there.
Many were calling for Ms Park's arrest, although a smaller number of her supporters also gathered in nearby streets.
There are fears the two sides may clash and there is a heavy police presence.
Currently, Moon Jae-in of the Democratic Party is leading in the polls, with one survey putting him almost 22% ahead of his nearest rival, acting president Hwang Kyo-ahn, who is loyal to Ms Park.
Mr Hwang has called for calm, saying the government should remain stable to prevent internal conflict from spreading.
However, police are bracing themselves for more violence following the deaths of two of Ms Park's supporters on Friday.
A third person, aged 74, is understood to have had a heart attack during Friday's protests, and died on Saturday, according to Reuters.
Ms Park's office said she would not be leaving the Blue House, South Korea's presidential palace, on Friday nor making any statement.
It is understood she will not leave until her own home in Seoul is repaired and cleaned.
Why did Park lose her job?
At the heart of the drama lies the close friendship between the president and Ms Choi.
Ms Choi is accused of using her presidential connections to pressure companies to give millions of dollars in donations to non-profit foundations she controlled.
Ms Park is alleged to have been personally involved in this, and to have given Ms Choi unacceptable levels of access to official documents.
Parliament voted to impeach Ms Park in December and the Constitutional Court has since been deciding whether to uphold or overturn this.
On Friday, a panel of eight judges ruled Ms Park's actions "seriously impaired the spirit of... democracy and the rule of law".
The court said she had broken the law by allowing Ms Choi to meddle in state affairs, and had breached guidelines on official secrets by leaking numerous documents.
Ms Park had "concealed completely Choi's meddling in state affairs and denied it whenever suspicions over the act emerged and even criticised those who raised the suspicions," it said.
But the judges dismissed some charges, including accusations Ms Park had infringed on freedom of the press by creating a media blacklist of cultural figures, and criticism of her response during the 2014 Sewol ferry disaster.