Police broke up their protest by using water cannon, in the first implementation in Cairo of powers given to them by the law on demonstrations enacted at the weekend.
Hoda El-Sadda, a senior member of the 50-member panel, told AFP she and nine others had "suspended work after the arrest of protesters".
As a result, the panel suspended its work for the day, but adjourned its session to Wednesday morning, television reports said.
Dozens of protesters were demonstrating in front of the Shura Council, where the constitution panel sits, against an article in the revised basic law that allows the military to try civilians in certain cases, a security official said, when police used water cannon and made arrests.
Among those arrested were Mona Seif, founder of a campaign against military trials of civilians, and Ahmad Harara, a dentist who lost his eyes to birdshot during protests against Hosni Mubarak in 2011 and against the military junta that ruled Egypt immediately after Mubarak's fall.
Another panel member, Khaled Youssef, also criticised authorities for arresting the protesters as he suspended his work on the committee.
"What is happening is threatening the future of our country because this committee of 50 faces risk of exploding. This committee is the cornerstone of the road map," Youssef said on state television, referring to the transitional plan outlined by Egypt's interim rulers after they came to power following the overthrow of Islamist president Mohamed Morsi.
The road map envisages that a new constitution, a new parliament and a new president should be in place by the summer of 2014.
"It was a peaceful demonstration and those who have been arrested are all known in Egypt. They were not violent... and we are here to put together a constitution that guarantees freedom to Egyptian people," Youssef said.
'Law must be repealed'
Activists vented their anger at Tuesday's crackdown.
"This won't stop the protest", said Alaa Abdel Fattah, a prominent activist and brother of Seif.
"They used water cannon and then started beating and arresting people," he told AFP.
Ahmed Maher, founder of the April 6 movement that spearheaded the 2011 revolt against Mubarak, said "the law must be repealed".
"The interior ministry does not want any protests," he said, adding that "even under Mubarak we could hold protests".
The ministry said it had told Maher to get permission before holding a separate demonstration which demanded the prosecution of those responsible for the deaths of demonstrators in November 2011 when opponents of the then ruling military junta clashed with police in Cairo.
"We told him but he refused and did not abide by the law. Police then dispersed the demonstration using water cannon," the ministry said, adding protesters pelted stones.
Police meanwhile fired tear gas on protesters as new protests were held in downtown Cairo, with demonstrators chanting slogans against the military and its chief General Abdel Fattah al-Sisi and interior minister Mohamed Ibrahim, an AFP reporter reported, adding that shopkeepers in the area closed their businesses.
Amnesty International said the new protest law was a "serious setback that poses a grave threat to freedom of assembly and gives security forces a free rein to use excessive force, including lethal, against demonstrators".
The Islamist Freedom and Justice Party, the political arm of the banned Muslim Brotherhood, also condemned the law.
It said that "instead of stopping the security forces' repressive and murderous practices, (the law) entrenches brutality and abuse".
Egypt's military-installed authorities are engaged in a brutal crackdown since August 14 on Islamist supporters of Morsi, the country's first democratically elected leader.
He was overthrown on July 3 after mass street protests against his turbulent single year of rule.
Since then, more than 1,000 pro-Morsi protesters have been killed and thousands more arrested.
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