Would an aircraft in such state maintain constant distance to
the earth i.e fly concentric to earth surface with constant
altitude or will it continue straight 'ahead' with slowly
In any flight random motions of the air will pitch (and roll and yaw) the airplane to a smaller or bigger extent. You can be flying in an updraft or downdraft. Flying the airplane is that you correct for those random jostlings. So the pilot (or autopilot) is always in the process of keeping the airplane level and at the designated altitude.
A side-effect of that is that the airplane keeps flying along a concentric circle around the Earth's center.
Following the Earth's curvature requires no separate input because the corrections for the random jostlings are far bigger anyway.
But yeah, as a thought experiment: in a perfectly motionless atmospere that would allow you to fly dead ahead you will over time gain altitude as the earth curves away from under the airplane.
The text in the above picture is confusing. To the left it says: 'no gravity'
What is intended, I guess:
On the left side: 'Resultant force of gravity and lift is zero'
Let me make a comparison with an orbiting satellite. In orbit there is only one factor: gravity. For a satellite the curve of the trajectory is due to gravity only. The satellite is falling down, but it has so much velocity component parallel to the Earth's surface that it never reaches the Earth.
For an airplane the altitude comes from the resultant force of gravity and lift. To fly along a circle that is concentric with the Earth the lift force must be slightly less than the gravity, in such a way that the resultant force of the two is just enough to keep the airplane moving along that circle with that velocity.
Imran Ullah has pointed out a subtlety, and I think I should address that.
The initial condition is that the direction of gravity force and the direction of lift force are exactly opposite.
Recapitulating: if the autopilot keeps the orientation of the aircraft the same then the aircraft proceeds on a straight line that is tangent to the surface of the Earth. As the aircraft moves along that line over time the direction of gravity force will no longer be exactly opposite the lift force. As indicated by Imran Ullah, the new resultant force will not be zero: it will point rearward, and slightly upward.
So, if the autopilot is programmed to maintain constant air speed, but to leave the altitude to whatever happens, the prediction is that the flight path will curve up.