Bleeding that occurs after an egg retrieval procedure – does it necessitate ghusl?

The bleeding that makes ghusl obligatory for a woman is menstrual bleeding and postpartum bleeding (nifas) only.

In your case, it seems that the bleeding only resulted from the needle used for retrieving the eggs, because retrieval of the eggs from the woman’s ovaries is done by inserting a needle into the ovary, as has been discussed previously in the answer to question no. 234124 .

What appears to be the case makes us think it most likely that this bleeding is not menses.

The rulings are to be based on what is most likely to be the case, when we cannot be certain.

Al-Qarafi (may Allah have mercy on him) said:

In principle according to Islamic teachings, nothing is to be taken into account except what we know for certain, because Allah, may He be exalted, says (interpretation of the meaning): {And do not pursue that of which you have no knowledge} [al-Isra’ 17:36]. That is because there is no room for error in that case, but in most cases it is difficult to have certain knowledge. Therefore Islam permitted following what is most likely to be the case, because it rarely turns out to be wrong, and most of the time it turns out to be correct. When there is doubt, the matter should be regarded as remaining as it originally was. So whenever there is doubt, the thing is not to be taken into consideration, and we should refer to what was originally the case before we had doubts about what it is now …"(Adh-Dhakhirah  1/218-219).

Based on that:

This blood is not menses, but it does invalidate wudu’. If it is ongoing, then it is istihadah (irregular bleeding).

For information on the rulings on istihadah, please see the answer to question no. 68810 .

And Allah knows best.