Halimah Yacob, a former speaker of parliament from the Malay Muslim minority, was sworn in Thursday without a vote after two potential rivals were disqualified from running.
The protest followed outrage on social media over what critics say was the engineered victory of an establishment figure.
While the role is largely ceremonial, the president has veto powers on key government appointments and safeguards the country's substantial financial reserves.
"We are not against (the president)... We are against the system. We are against the process against the right to exercise our democratic vote," Goh told media after the protest.
Scores of Singaporeans dressed in black milled around the park, some holding up signs calling the presidency a "(S)election", while others wore T-shirts emblazoned with the hashtag "not my president", which first gained popularity among Americans unhappy with the victory of Donald Trump.
Goh said an earlier rally was cancelled because a special police permit was needed since the event would have touched on race.
Sedition laws in Singapore make it an offence to promote hostility between different races or classes in the multiracial society, which is mainly ethnic Chinese with large Malay and Indian minorities, but critics say this stifles free speech.
Authorities decided to allow only candidates from the Malay community in the presidential election to foster harmony in the city-state of 5.5 million people which is dominated by ethnic Chinese, and give more opportunities to minorities.
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