An appeals court in February last year upheld a 15-year jail term against Abulkhair on a series of charges including "inciting public opinion".
"The prison administration has not allowed him to take medical tests, prevented him from ordering appropriate foods that suit his health condition, and continues to prevent him from special visits or reading books or daily newspapers," the Gulf Centre for Human Rights said.
Abulkhair started his hunger strike on Tuesday at his prison in the Red Sea city of Jeddah, said the rights group which has offices in Beirut and Copenhagen.
It condemned prison administrators for "targeting" Abulkhair and called for his immediate release.
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Abulkhair last year won the Ludovic Trarieux International Human Rights Prize, awarded by European bar associations to lawyers who defend human rights and fight intolerance and racism.
South African anti-apartheid icon Nelson Mandela was the first recipient of the prize.
Abulkhair defended Saudi blogger Raif Badawi, who was arrested in 2012 and convicted of insulting Islam.
Badawi is serving 10 years in jail and has received 50 lashes of a 1,000-lash sentence that caused international outrage.
The two activists were nominated for last year's Nobel Peace Prize by a Norwegian member of parliament.