The force said it estimated there were up to 550 people across the two protests.
It has issued a face covering ban and a dispersal order until 20:00 GMT for part of the town, allowing officers to tackle anti-social behaviour.
The anti-migrant protesters marched from the Haslar Immigration Removal Centre (IRC), along the Haslar Road and into the town square along a designated route.
The Unity protest was static and at the far end of the high street.
The IRC is being redeveloped to house up to 130 migrants with no lawful right to remain in the UK before they are deported.
The site has been closed since 2015, with redevelopment plans announced by the Conservative government in 2022.
Earlier this year, the current government announced plans for an additional 470 beds at the site, beyond the original proposal for 130.
Almost 500 comments have been submitted on the planning application, the majority opposing the expansion.
Gosport has seen several anti-migrant protests in recent weeks, including demonstrations in September and October following the arrival of small boat migrants at Haslar Marina and Fort Blockhouse.