How Theory of General Relativity Augments/Replaces Galilean Inertia and Newton's First Law of Motion
An Object is said to be in a State of Galilean Inertia if it is either in a State of Rest or State of Motion with Uniform Velocity in a Straight Line. Any Observer in such a a State is called an Inertial Observer.
The Theory of General Relativity Augments / Replaces the definition of Inertia via the Principle of Equivalance which states that
The Absence of a Gravitational Field (i.e. True Weightlessness) is indistinguishable (i.e. no such experiments can be designed to find out the difference) from Free Fall Under Acceleration in a Gravitational Field (i.e. Apparent Weightlessness).
The Local effects of Gravity (for e.g weight) are the same as those of being in an Accelerating Reference Frame. That is, Weight observed due to Accelerated Motion in the Absence of a Gravitational Field (i.e. Apparent Weight) is indistinguishable from Weight observed in State of Rest or Uniform Linear Motion in the presence of a Gravitational Field (i.e. True Weight).
Therefore as per Principle of Equivalance, any Object in any of the above scenarios can be said to be in a State of Inertia. Hence, any Observer in any of the scenarios above can be called an Intertial Observer.
As per Newton's First Law of Motion all Objects maintain their State of Inertia (i.e. State of Rest or State of Motion with Uniform Velocity in a Straight Line) unless they are acted upon by a Net Non-Zero External Force.
The Theory of General Relativity Augments / Replaces the Newton's First Law of Motion as follows
Since Forces experienced due to one or more Gravitational Fields cannot be entirely abolished, any Object/Body stays or remains in a Motion of Free Fall Acceleration under one or more Gravitational Fields
tracing a Geodesic Path (i.e. a Path of least action/length in Space-Time) unless acted upon by a Net Non-Zero External Force. The actual nature/trajectory of the Geodesic Path is determined by the Curvature of the Space-Time.