Some quick background
Pixels are the smallest physical "dots" that are lit up on the monitor to display an image.
They are the building blocks and define all of the tradeoffs. The monitor is manufactured with a specific arrangement of pixels, which is its "native" resolution.
Characters are drawn on the screen by defining which pixels are illuminated within an imaginary grid. The number of pixels in the grid determines the size of the font on that monitor.
Let's start with screen fonts at their normal size and the computer configured to use the monitor's native resolution, and compare how the same font will look on two different size monitors. On each monitor, the actual size of the font on the screen will be determined by the physical size of the screen's pixels.
The density or closeness of the pixels at which the screen is manufactured is measured in pixels per inch. That determines the physical size of each pixel. A 19" monitor with a native resolution of 1600x900 pixels, and a 23" monitor with a native resolution of 1920x1080 pixels, both have roughly 96 pixels per inch. So, if these two screens were compared side-by-side, the font would be the same size on both displays.
If you want to select a larger font or set the computer to magnify the font, either option reduces how much will fit on the screen. The larger screen will give you more screen real estate (more pixels to work with). So, the larger, higher resolution monitor will allow you to display more of the enlarged content on the screen, offsetting the content loss from enlargement.
If you set the computer to a lower resolution and magnify it to fill the screen, it maps the content of the smaller image onto the larger space and interpolates to determine what each physical pixel displays. Mapping a 1280x720 resolution onto a 1600x900 display is equivalent to a magnification of 125%. If you wanted that same magnification on a 1920x1080 display, you would select a resolution of 1536x864 (or whatever was the closest standard resolution available), to map full screen. That figure is nearly the same as the native resolution of your current monitor. If you selected a slightly higher resolution, you would get a slightly lower magnification.
So with the larger monitor, you could come close to displaying the content of your current monitor's native resolution at the magnification you like.
Hstoerr raises a good point in his comment. The question talks about crisp, clear text at the monitor's native resolution. Displaying content at the content's native resolution will be sharp. Using any form of magnification, or mapping lower resolution onto a higher-resolution screen, will lose that sharpness. The process of interpolating and averaging pixels degrades edges and fine detail. The sharpest results will be obtained by using larger native content (for example, large fonts and icons displayed without magnification).