One of the demonstrations, held by nationalist groups the National Movement, All-Poland Youth and National-Radical Camp, urged the president to sign the disputed new law with banners which read: “Take off your skullcap and sign the bill” and “Be brave, Poland, president, sign the bill”.
Two of those groups have in the past been accused by critics of staging racist and fascist marches.
In a counter-protest, members of the left-leaning Polish Citizens group carried white roses and chanted: “We don't condone antisemitism”.
Both houses of Polish parliament have passed a bill that would introduce penalties of fines or up to three years in jail for anyone who publicly ascribes blame or joint blame to the Polish nation or state for crimes committed by Nazi Germany or for war crimes or other crimes against humanity.
The bill has ignited tensions with Israel, which could amend its own laws to have Poland accused of Holocaust denial, according to the Jerusalem Post.
Israeli ambassador to Poland Anna Azari said: "In Israel, this bill is seen as creating a possibility of punishment for Holocaust survivors' testimony”.
The bill has also been criticised by the US State Department, which said: “We encourage Poland to reevaluate the legislation in light of its potential impact on the principle of free speech and on our ability to be effective partners.”
The bill was protested in Ukraine because it could see penalties for anyone who denies crimes committed by Ukrainian nationalists until 1950.
In Poland, the planned new law is seen as a way of fighting the use of the phrase “Polish death camps”, which implies Poland's involvement in the Holocaust.
Polish government spokeswoman Joanna Kopcińska has said: “It was the Germans who attacked Poland, while the Poles and Jews were the victims”.
“There were no Polish death camps, no Polish concentration camps or Polish extermination camps. We must set the record straight by continually explaining and clarifying things,” she added.
GermanDeathCamps.info, a new educational website aimed at debunking misconceptions about Poland’s role in the Holocaust, has been launched by Polish Radio.
The Polish bill needs to be signed by President Andrzej Duda before it enters into law. (vb)