There is no reason to think that white Southerners would have consented to end segregation anytime in the short term. They might have consented in the long term, but not sooner.
We can look at the track record of the South to show that they had no real desire to end segregation. Perhaps the most notable example of this was Governor George Wallace who promised "segregation now, segregation tomorrow, segregation forever." Clearly, this was not a leader who was going to push for desegregation.
In the long term, economic pressures might have forced the South to desegregate. Companies might have refused to move to the South until they gave equal rights to blacks. There might have been boycotts against Southern companies. But this would likely have been similar to the move to end apartheid in South Africa and would have taken long periods of outside pressure to accomplish.