Two suspects were being interrogated by federal investigators on Wednesday, while state police were on guard against any unrest in the northwestern state.
"We are under strict orders to prevent any form of protests or demonstrations scheduled to condemn the murder," Hawa Singh Ghumaria, a senior police officer in Rajasthan, told Reuters, adding that the crime had sent "shockwaves through the country".
Brandishing a meat cleaver, two bearded men said in the video that they were avenging an insult to Prophet Mohammad caused by the victim.
They also alluded to Nupur Sharma, a former spokeswoman for the ruling Hindu nationalist Bharatiya Janata Party (BJP), whose remarks about the Prophet earlier this month triggered domestic and international outrage.
India's Home Minister Amit Shah said in a tweet that federal police had taken over the investigation into "the brutal murder" of Kanhaiya Lal Teli, giving the victim's full name.
"The involvement of any organisation and international links will be thoroughly investigated," Shah said.
Late on Wednesday, a spokesperson from Pakistan's foreign ministry rejected reports appearing in some Indian media that linked the suspects to a Pakistan-based organisation.
The two assailants had slashed Teli's head and throat in an attack while the tailor was taking measurements, according to Bhawarlal Thoda, a city administrator in Udaipur.
According to Thoda, the tailor had been detained over a social media post in support of the BJP spokeswoman that was traced to his mobile telephone, and that after being released Teli had told police on June 15 that he was being threatened by some group.
"Terrorists executed my father in the most shocking way, the country must stand with our family to demand justice," the victim's son, Yash, told Reuters after his father's body was cremated on Wednesday.
He said the culprits should be tried and sentenced to death, and denied that his father has made any remarks that would be offensive to other religions.
Politicians and prominent Islamic preachers condemned the killing.
"The incident has shocked followers of Islam, the heinous act committed by two men is absolutely un-Islamic," said Maulana Ahmed Siddiqui, a Muslim cleric based in Udaipur.
THREAT TO MODI
Authorities said they had suspended internet services in several parts of Rajasthan to prevent circulation of the video.
"The mood is tense and almost all shops are closed today," Thoda said. The city of around half a million people is one of the major tourist draws in the desert state, and is known for its luxurious hotels, including the famous Taj Lake Palace.
In another video clip posted online, one of the assailants also went on to threaten Prime Minister Narendra Modi, saying their blade would find him too.
Hardline Hindu organisations held protests in capital New Delhi to condemn the killing of Teli and some other protests were planned on Thursday.
India has a long history of religious violence, and thousands of people have been killed since the country became independent from British colonial rule in 1947.
Modi's pursuit of a "Hindu first" agenda since coming to power in 2014 has stoked tensions in a country where Muslims account for around 13% of its 1.4 billion people.
Earlier this month the BJP suspended Sharma from the party and expelled another official but the furore has not died down.
Prime Minister Modi has not commented on the incident in Udaipur. But, former Rajasthan chief minister Vasundhara Raje, who belongs to the BJP, blamed the Congress Party, now running the state, for the "communal frenzy and violence" that has arisen there.
Raje said "such acts can occur because the state government provides tacit support to criminals."
While Congress has championed secular values in India since independence, the BJP has cast it as a pro-Muslim party in order to draw Hindus away from its main opposition.
Rajasthan, with a population of around 69 million people, is just one of two Indian states where Congress holds a majority in the state legislature and it is due to hold elections next year.
(Reporting by Rupam Jain; Editing by Simon Cameron-Moore and Alex Richardson)
By Rupam Jain