My mother is a Indian citizen and will undergo major surgery in India. Can I claim this expenses under the US Tax?
If yes then do I have to pay bills with my US bank account or Debit or credit card?
Unfortunately, it is not likely your mother's expenses will qualify unless she is a US Resident. Also, she must be your dependent on your US Income tax return. How you pay for the expenses doesn't matter. Per IRS Publication 502 see below.
You can include medical expenses you paid for your dependent. For you to include these expenses, the person must have been your dependent either at the time the medical services were provided or at the time you paid the expenses. A person generally qualifies as your dependent for purposes of the medical expense deduction if both of the following requirements are met.
The person was a U.S. citizen or national or a resident of the United States, Canada, or Mexico. If your qualifying child was adopted, see Exception for adopted child , later.
You can include medical expenses you paid for an individual that would have been your dependent except that:
He or she received gross income of $4,150 or more in 2018;
He or she filed a joint return for 2018; or
You, or your spouse if filing jointly, could be claimed as a dependent on someone else's 2018 return.