At an infinitesimal point yes but across a volume tidal forces exist. There would be a slight stretching in the direction of the Sun and squeezing tangential to the Sun.

Across a year those forces would rotate 360° and not uniformly (we could calculate the eccentricity of the elliptical orbit from that).

Edit: measuring all that would be a lot more difficult because the earth is spinning so would have to figure that out so we can subtract the effects of that on our measurements.

It's hard to imagine how any species could properly understand physics well enough to do this without seeing the night sky so perhaps a better mind experiment is if a scientist got teleported to a sealed windowless box on a random planet somewhere in the universe could they tell by measurement whether the planet was in orbit around a star or not and the details of that orbit. I'm 99% sure they could from tidal forces and possibly by other means too.

Edit 2: another way to tell would be from the time-dilation differential from one side of the room to the other. A clock slightly deeper in the star's gravity well would run slightly slower.