Text won't become bigger with the resolution. Unlike movies and pictures. Things like text and web pages would appear the same size on a 17" or a 24" but the 24" would have more room to put more text and web pages.
You should look at pixel pitch. Most monitors have 0.25mm per pixel. They don't make monitors with anything larger than that. LCD TVs can have pixel pitch might higher though if you want bigger text.
Or you can just increase the zoom level on your web browser. CTRL + Scroll Wheel is the shortcut for increase/decrease zoom.
Generally speaking text on LCD screen will be pretty sharp as long as you use native resolution. Lowering the resolution will increase the size of the text, but will also make it a bit fuzzy due to interpolation. The lower the resolution from native, the less sharp it will be. Some monitors do have a sharpness adjustment in the onscreen menu if you use less than native resolution.
The ease of reading text on a 1920 x 1080 resolution monitor depends on the size of the monitor (or HDTV) and how far you are sitting away from it. The text on a 27" monitor with 1920 x 1080 resolution would be larger than on a 22" monitor with 1920 x 1080 resolution. It also depends on your vision.