Structurally, if my math is correct, no, it cannot hold together if made of concrete.
We can consider the ring as being analogous to a thin-walled cylindrical pressure vessel with a negative pressure. We start by describing the pressure:
$$P = F/A$$
Here, $A$ is the area of a given portion of the ring and $F$ is the force on the area. The force is equal to $\text{ gravity} \times \text{ density} \times \text{ volume}$, where the volume is the area of the portion of the ring times its thickness, or
$$F = g \times \text{ density} \times V = 9.8 \times 2400 \times A \times t$$
$t$ in the above equation is the thickness of the ring. This can be plugged into our equation for pressure to yield the following value after cancelling out $A$ (area) in the numerator and denominator:
$$P = 9.8 \times 2400 \times t$$
The tensile force for a cylindrical pressure vessel is $\frac{Pr}{t}$, although we have a negative force since gravity is compressing our pressure vessel, so we'll have an equivalent compressive force on our ring. We can plug our pressure equation in to get the following, after cancelling out thickness:
$$\text{ stress} = \frac{Pr}{t} = 9.8 \times 2400 \times r$$
$r$ in this case is the radius of the arch, which is roughly equal to the radius of the earth, or around $6370 \text{ km}$. We want this in meters to get stress in $\text{ MPa}$, so we'll convert to $2400 \times 9.8 \times 6.37 \times 10^6$, which comes out to around a total of $150 \text{ GPa}$, which is far greater than the compressive strength of concrete, which is around $800 \text{ MPa}$ for ultra-high performance concrete, and also much higher than the strength of materials such as steel or quartz. It's on the same order of magnitude as the compressive strength of diamond, but diamond is about 50% denser than concrete, so it would still probably fail. The stress is around the same point as the maximum predicted stress of nanodiamond, but this hasn't been tested in a lab.
Values used for calculations:
- Acceleration due to gravity of $9.8 \text{ m/s}^2$
- Concrete density of $2400 \text{ g/m}^3$
- Radius of $6370 \text{ km}$
- Maximum compressive stress of $800 \text{ MPa}$
All of this is under the assumption that the ring is relatively stationary with respect to the earth.