In a carefully worded statement, the president's office wished "Russia and the Russian people ... success in the political, democratic, economic and social modernisation of the country" following Putin's landslide win in Sunday's vote.
But, while calling for "a constructive dialogue between Russia, France and Europe", Macron took a tough line on the poisoning of former spy Sergei Skripal, and his daughter, Yulia.
"The president called on Russian authorities to shed light on the responsibilities for the unacceptable attack in Salisbury and to firmly regain control of any programmes that have not been declared to the Organisation for the Prohibition of Chemical Weapons," the statement said.
"He informed the Russian president of measures taken to ensure the security of our citizens," it added, without going into details.
Macron also expressed "great concern" over the situation in Afrin, which was captured from Kurdish militias by Turkish-led fighters Sunday, and in Eastern Ghouta, which has been systematically bombarded by the forces of Russian ally Bashar al-Assad.
National Front hails "stability"
The far-right National Front (FN) was more upbeat in its response to Putin's reelection.
"This great electoral success confirms the stability and democratic consolidation of Russia," a party statement said. "It is evidence of the wishes of the Russian people to continue on the road of reform started by President Putin, which have notably led to the return of this major power to the international stage in recent years."
The FN called on the EU to revise its policy in relation to Russia and drop its "absurd and counterproductive police of blackmail, threats and sanctions".
FN leader Marine Le Pen visited Putin in March 2017 and the party received a loan from a Russian bank in 2014.
Le Pen's partner, Louis Aliot, who is a French MP, was in Moscow on Sunday, invited to observe the election by the Russian authorities.