This is not altogether correct. In particular, it does not matter what the pressure of the room is (unless you reach extreme values), because actually what is holding the water column is not some "high" value of the pressure at the bottom of it, it is the fact that the pressure at the top of the water column (at the water-finger interface, or in the air pocket between water and finger) is lower than the atmospheric pressure. It is exactly the room pressure minus $\rho g h$, where $\rho$ is the density of water, $g$ gravitational constant and $h$ the height of the column.
This is true whatever the radius of the straw. However, you need two more ingredients to keep the system stable :
First, the air pocket should remain incompressible : this is true for small enough water columns, but there'll be some height $h$ above which it will expand under the weight of water. And even if there is no air, if you get to pressures lower than 0 at the top, the phenomenon of cavitation will nucleate a bubble of void that will expand.
Second, if the straw is too wide, the bottom surface will grow unstable and water will flow e.g. on one side, allowing bubbles to rise on the other. What actually maintains the stability there is a force called surface tension. You can lower it by adding a surfactant (dish washing soap e.g.) to the water, you'll see that the biggest straw possible is then smaller!