Overall, this seems to be a question of trade-offs. The better the resolution, the slower the page load for a do-nothing page. Is it worth the slow-down to have the better resolution and avoid pixellation?
Also, I think you're asking the wrong question, since a 24-inch screen can be in multiple resolutions.
I would approach this in the following manner:
what is the largest resolution you MUST have your photo look "good" on? Then make your photo that resolution. If 24" is your target, look at what resolutions this size monitor "typically" supports and target that.
What number of colors you want? (or perhaps b&w / grayscale). If you reduce the number of colors (preferably to "web-safe" colors), you can load faster with the same resolution.
A program like Photoshop (or Gimp) will probably give you the most power in tuning these parameters.
Do you care if only a portion of the photo displays when your viewer has a smaller window?
I know this isn't a cut and dried answer, but these things seldom are (IMHO).