Dozens of demonstrations were held in France including in Paris, as well as in Brussels, Berlin, Madrid and Palermo.
Many dressed in the boat's orange colours and chanted for “a flag for Aquarius” while slamming the “criminal inaction” of EU governments to resolve the impasse.
Marseille, home to the SOS Mediterranee NGO which operates the vessel with Doctors Without Borders (MSF), saw the largest show of support. Organisers said 10,000 people took part in the demonstration though police put the figure at nearer 3,500.
A similar rally in Paris also drew a crowd of around 10,000, according to the organisers, while police put the figure at a tenth as many.
“The goal is to make an appeal to citizens and show that SOS Mediterranee draws its legitimacy from civil society,” said the NGO's spokeswoman Sophie Rahal in Paris.
Panama last month pulled the ship's flag following a request from Italy's far-right, anti-establishment government, meaning the Aquarius cannot legally set sail.
The boat's operators say it has rescued some 29,500 migrants to date and SOS Mediterranee president Francis Vallat said “the situation is more critical than ever”.
Vallat said although migrant volumes had dropped “we are passing from one death per 42 migrants (at sea) last year to one in 18 migrants” this summer.
Attending the Marseille rally, French leftist champion, local lawmaker and leader of La France Insoumise Jean-Luc Melenchon insisted “Aquarius must have a flag — and a French flag”.
Meanwhile, around 100 flouted an official protest ban and gathered in the northern port of Calais — where authorities in late 2016 dismantled the notorious Jungle refugee camp, which at its height at been home to some 10,000 migrants.
Immigration remains a hot-button issue in the European Union following the 2015 migration crisis. There are deep divisions among member states over how to handle the continent's biggest influx since World War 2.
Frontline state Italy has clamped down on asylum-seekers under far-right Interior Minister Matteo Salvini.
Saturday saw protesters gather in Riace — a small hamlet in Italy's deep south which has earned praise for its integration of migrants — demanding the release of their leftist mayor.
Domenico Lucano was placed under house arrest on Tuesday under suspicion of arranging “marriages of convenience” for immigrants and for allegedly skipping a tender process while awarding a garbage management contract to co-operatives with ties to migrants.
In office since 2004, Lucano has made headlines around the world for welcoming migrants to the sparsely-populated village in Calabria in a bid to boost jobs and development.
“Keep strong, continue fighting,” his supporters chanted Saturday as they gathered outside his window from where he raised a fist in support.