The placard with an image of the Kremlin critic reads "Freedom to Navalny!"
Navalny, who the West says has been wrongly jailed and should be freed, returned to Russian Federation in January after recovering from what German doctors say was a nerve agent poisoning.
Navalny is Russian President Vladimir Putin's most visible and adamant opponent.
Asked about Navalny's worsening condition, U.S. President Joe Biden told reporters Saturday: "It's totally totally unfair and totally inappropriate".
Mr Navalny said on Friday that prison authorities had threatened to put him in a straightjacket to force-feed him unless he abandoned his hunger strike.
Prison authorities say they have offered Navalny proper medical care but that the 44-year-old opposition politician has refused it and insisted on being treated by a doctor of his choice from outside the facility, a request they have declined.
The 44-year-old began a hunger strike on March 31 to demand proper medical treatment for back pain and numbness in his legs and hands.
Navalny's wife Yulia said her husband now weighed 76kg (168 pounds), which is down 9kg since starting his hunger strike, according to AP.
Having blood potassium levels higher than 6.0 mmol (millimole) per liter usually requires immediate treatment.
The medics said he had to be examined immediately "taking into account the blood tests and his recent poisoning".
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Navalny's spokeswoman Kira Yarmysh said on Saturday that the situation was critical.
"With his condition it's a matter of days".
"Allow a doctor to see my dad", Navalny's daughter Dasha, a student at Stanford University, wrote on Twitter.
The open letter was published in Le Monde newspaper in France, The Economist, La Repubblica in Italy and Der Spiegel.
In early April, as NPR previously reported, Navalny said he had developed a temperature and a cough.
"Putin only reacts to mass street protests", she added.
Navalny's allies said they would set a date for the protest once 500,000 supporters had registered with a website.
The Russian Public Prosecution demanded Friday that the network of regional offices of Navalny and his anti-corruption institution be classified as "extremist organizations". Police arrested more than 10,000 people during nationwide protests in January calling for Navalny to be freed.
"The darkest times are beginning for free-thinking people, for civil society in Russian Federation", said Leonid Volkov, the head of Navalny's regional offices.