Earth will become a barren rock.
Assuming gravity works by line of sight (if two point masses $m_1$ and $m_2$ are on opposite sides of the portal, and there is a line segment through the portal that connects them, each will experience gravity as if, instead of going through the portal, the line segment continued going through space and the other mass was on the other end of the line segment). This gets rid of some of the discontinuities in the strength of gravity that would happen if just the field lines passed through the portal.
Then, using the equations $\color{#328396}{a=\frac{-1}{\rho}\frac{dP}{dz}}$ and $PV=nRT$, we get $a=\frac{-kT}{P}\frac{dP}{dz}$. Assuming that some equilibrium is reached, $a_{net} = 0$, so $a_g=\frac{kT}{P}\frac{dP}{dz}$ for some positive constant $k$.
Next, assume that when equilibrium is reached, the air pressure on earth is non-zero, and consider the line that if you were on and looked through the portal, you would be looking straight at the sky, directly away from Earth. From that line, none of Earth is in view, so nothing on that line will experience gravity. Therefore, on that line $a_g=0$ meaning that $\frac{dP}{dz} = 0$. Also, at any point next to the portal, the pressure will be equal to the pressure where the portal is on Earth. That means that on our line, the pressure at all points must be the same pressure as on earth's surface. However, this is impossible. Because the line extends to infinity, it would require an infinite amount of gas to maintain a positive pressure on the whole line. This contradiction means that the pressure on Earth's surface must be 0.
In reality, the gas that passed through the portal would eventually have enough mass that its gravity would be significant, preventing the contradiction above from arising. However, because Earth has so little gas (only about $5*10^{18}$kg), the pressure on Earth's surface would likely be very very approximately a millionth of an atmosphere once equilibrium is reached.