Private gynaecologist for child birth

Child birth is normally only possible in a hospital (and costs €5000-10000 if there are no complications. This would be fully covered by your public health insurance.
Other possibilities like home birth are only possible with prior arrangements. The cost varies, but is certainly much higher than the standard hospital birth above (as are the associated risks). If there were any complications, they'd send you to the public hospital nevertheless.

What if one wants to have child birth in a private hospital?How much would it cost....(just an estimate)?

I don't know of any private hospital in  Germany. If there are any, they are very few and a reserve of the super-rich (with prices to match - it would certainly be a multiple of the public hospital costs I mentioned above).
But you can enter any public hospital as a private patient, which costs more than as publicly insured person and gives you more luxuries (e.g. a single room and more visits by the head physician). In that case you pay only the difference between the (insurance-covered) standard treatment and whatever you book.

Thanks ......!!!

I just googled private hospitals in Stuttgart and found that there are actually a few, but they specialize in treatments that public hospitals don't offer, like plastic surgery or abortion.
There is (or rather, was, since they close down in December) a semi-private birthing clinic associated with the (public) Robert-Bosch-Krankenhaus. It's called Charlottenhaus and my wife considered it for our second child. We decided otherwise in the end, not because of the cost (which was affordable due to the abovementioned differential arrangement), but because she needed a Caesarian and felt more comfortable having that in a proper hospital.

I think the real question is what you really want and what you even mean by a private doctor. Do you mean a doctor with a private practice rather than working in a hospital because there are plenty of these? But a doctor even in a hospital can have private patients and like Beppi mentioned, private clinics and hospitals are the exception. I can imagine that coming from India one has experience of public hospitals that lack privacy and a high level of care. This is not relevant in Germany. One gets the very best of care even in a normal public hospital in Germany. For extra privacy what might come in question is having a private room rather than having to share. One should inquire at local hospitals offering birthing services (not all do) about what you would have to pay in addition since it is not covered by public health insurance.

Thanks for the reply...!

Alright..