"Yousuf's wife has undergone some complications in her pregnancy and is due to undergo surgery on January 12," Pakistan Cricket Board director of communications, Ahsan Malik, told AFP on Monday.
Doctors have said the operation may be delayed until January 19.
"If the operation is held on January 12, he would fly to South Africa on the 14th, but if it happens on January 19 he will miss the second Test as well," Malik added.
Pakistan start the first Test against South Africa at Centurion from January 11, and Yousuf's absence will be a blow to a team already hit by injuries to fast bowler Umar Gul and allrounder Shoaib Malik -- both suffering from ankle problems.
The second Test is scheduled for Port Elizabeth from January 19. The third Test starts in Cape Town from January 26.
Bowling coach Waqar Younis also refused to go on tour and resigned in protest over his appointment, which was only for the Tests and not for the five-match, one-day series which follows.
Yousuf, who created a world record for the most runs in a calendar year with 1,778 in 2006, was due to leave for South Africa before the first Test.
The 32-year-old, whose father died last month, has been the mainstay of Pakistan batting, having scored 6,402 runs in 73 Tests and 7,608 in 228 one-day internationals.