A statement carried on state media said Doha would "not take any measures against residents of Qatar who hold the nationalities of countries that severed diplomatic ties or lowered diplomatic representations with the state of Qatar, on the back of hostile and tendentious campaigns against the country".
It said Qatar was acting in "accordance with its firm beliefs and principles".
The decision will affect more than 11,000 people from the Gulf states of Saudi Arabia, the United Arab Emirates and Bahrain alone, according to official figures.
Saudi Arabia and allies including the UAE and Bahrain cut diplomatic ties with Qatar on Monday, accusing it of supporting terrorism.
As well as cutting air, sea and land links with Qatar, the countries ordered its citizens to leave within 14 days.
'For potentially thousands of people across the Gulf, the effect of the steps imposed in the wake of this political dispute is suffering, heartbreak and fear," said the human rights group has claimed.
Figures from Doha's National Human Rights Committee show that 8,254 Saudi residents live in Qatar.
There are 2,349 Bahrainis and 784 Emiratis in the country.