No, Special Relativity showed that there is no such thing as a standard reference frame.
The description of motion requires what is called a inertial reference frame, the presumed point from which motion is described. To say that the earth revolves around the sun is a statement that the earth’s motion is relative to the sun (hence the term ‘Relativity’)
What Einstein showed in Special Relativity is that there is no such thing as an objective standard refernce frame.
Thus, when describing motion, it is equally valid to use either point ( sun or earth) as the inertial reference frame.
en.wikipedia.org/wiki/Special_relativity#Lack_of_an_absolute_reference_frame
What Galileo did was to change what was then the internal reference frame from the Earth ( all motion was described relative to the Earth) to the Sun (all motion was described relative to the Sun). But Einstein show that there really is no true inertial reference frame, only relative one (again, that’s why Relativity is called ‘Relativity’ :

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