you pay income tax in the country you live, not where you work (i.e. if you were employed by a US company, but working remotely in the UK, you'd pay income tax in the UK)
Any country can try to extract any tax from anyone it likes, and in general the US does.
Except if you are working for a company that is sending you to the US (your company makes the income). It's easy to argue for 2-3 weeks, harder to argue for over 2 months.