Do Mathematics Grad schools in the US expect a research paper from a Masters Student for admission into a PhD Program? I have recently joined a Masters program in Pure Mathematics in an Asian University.

My course does not have a research component, only coursework like - Field Theory, Algebraic Topology, Topology-I,II, Differential geometry, Module Theory, Fourier Analysis, Computational Differential Equations, Measure Theory , Complex Analysis, Algebraic Number Theory, Homology Theory etc.

Though, I would love to do research and publish a paper, however, some of the topics above would be new to me. So, I am not sure, if I would even get time to do research and do well in the coursework as well.

I plan to apply to the US for a PhD after my masters. So, would the school expect a research publication towards the end of my masters? Will not doing so affect my application for a PhD?
As mentioned in the comments, no.  In the US, there are typically 2 master's tracks one can take: a thesis option (with some coursework) or a non-thesis option (more coursework + exams).  Most students I know who do master's degrees in the US (many of whome go on to PhD programs) do the non-thesis option.  In fact, many undergrads do a non-thesis masters (or something like Cambridge's Part III) to help them get into good PhD programs.  So in the US, in contrast to some countries like Japan, it is not expected.