Several hundred people were marching from the Delmas to Petionville neighborhoods when shots were fired from nearby.
It's believed the injured were two protesters, a police officer and a journalist.
Their condition wasn't immediately known.
Moïse said earlier Monday he would not step down and offered to continue to try to negotiate with the opposition.
He was speaking at a ceremony commemorating a key battle of the Haitian Revolution.
Moise is under pressure to quit from critics who accuse him of economic mismanagement and a failure to investigate corruption.
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