The advert by TRESemmé, a Unilever Plc brand, showed an image of African black hair which it described as "dry and damaged", while an example of white hair was referred to as "fine and flat".
The advert, posted on Clicks' website on Friday, provoked a backlash on social media and the retailer removed it the same day. It apologized, as did TRESemmé South Africa.
But on Monday, protesters led by hard-left party Economic Freedom Fighters (EFF) demanded Clicks stores be shut for at least a week. Some wore EFF's trademark red hard-hats and overalls and sung struggle songs outside Clicks stores.
"White people insult us and then they apologize, they think that's the end. We are no longer going to accept any apology which is not accompanied by justice," EFF leader Julius Malema told supporters outside a closed Clicks store in Polokwane in the province of Limpopo. "Who is punished for projecting black people as ugly people?"
Video footage on Twitter showed protesters toppling shelves and destroying products in one store. Another, in Witbank in Mpumalanga province, appeared to have suffered fire damage.
Clicks said there were protests at 37 stores in Kwa-Zulu Natal, Gauteng and Western Cape provinces. Seven stores were damaged. It was not clear which stores had shut.
"The negligent employees have been suspended, and we have engaged the supplier, who has now also issued an apology," Clicks Chief Executive Vikesh Ramsunder said of the advert.
Facing accusations of racial prejudice, Unilever said in June it would drop the word "fair" from its "Fair & Lovely" skin lightening products.
Many makers of consumer packaged goods have reconsidered their marketing following global protests against racial injustice. Several brands have scrapped Black advertising mascots.
(Reporting by Nqobile Dludla and Mfuneko Toyana; Editing by Promit Mukherjee and Timothy Heritage)