For foreigners with no US source income, having to pay taxes is based principally on residency which is a defined term. If you are resident, you pay. If not, you don't.

Credit card companies generally do not issue cards to non-resident foreigners (for obvious reasons).

So it's difficult to see how the OP can persuade the cc company to issue him or her a card, but at the same time persuade the IRS that he or she is not resident for tax purposes. To do so would involve lying on one or more occasions. This would be more than unwise as committing fraud (as it would be in lying to the cc company) could preclude all sorts of things.