Is it permissible for a mother to pray after her private parts have been touched by a midwife for examination, without performing ghusl?

The sick person is not required to perform ghusl (ritual bath) if the midwife touches her private parts or inserts her hand therein, because there is nothing that necessitates ghusl.

As for invalidating wudu (ablution):

Touching through a barrier: Does not invalidate wudu, even if accompanied by desire, because actual physical contact did not occur. Touching without a barrier: Does not invalidate the wudu of the touched person (the patient), unless desire occurs. Invalidates the wudu of the toucher (the midwife) if the touch is without a barrier. Emission of something from the two private parts: If the midwife withdraws her hand and it is stained with blood, yellowish discharge, or moisture, then the patient's wudu is invalidated; because everything that exits from the two private parts invalidates wudu.

In summary: The touching of the patient's private parts by a doctor or midwife does not invalidate wudu except in two cases:

1. Something comes out from the patient with her hand (urine, blood, etc.). 2. The touching is without a barrier, and is accompanied by desire.