Pretty much any modern DSLR will do good product shots of watches and such - in fact the kit lens that comes with most DSLRs will be more than suitable for that kind of work since the majority of the quality in those kinds of shots will come from the correct control over the lighting of the subject itself.
If you're just after product shots for web display then you can easily get by with some lesser gear and just good lighting control - if you want higher quality then a DSLR can offer you that for a higher price of course.
A few thoughts:
1) A DSLR from Canon, Nikon, Pentax, Sony etc.. will do this job and do it well.
2) If you have more budget a macro lens (60mm to 90mm is what I would suggest for indoor product photography) can provide the ability to focus closer for smaller objects whilst delivering a high quality result
If you are on a more strict budget then extension tubes and the kit lens with a DSLR will work well
3) Lighting is KEY - you don't need complex or expensive gear (you can and of course higher grade items will help) but you do need correct control over it.
If you go the DSLR path I'd suggest the following books:
Understanding Exposure by Bryan Peterson
The Digital Photography Book (books 1 to 4) by Scot Kelby
These will help you a lot in learning how to control and use the camera
Whatever option you go for I'd suggest the book:
Light Science and Magic (version/edition 4 is the latest).
This will help you a lot in learning about lighting; how to light and such.
If you have a rough budget in mind (and can list your country so we know what kind of prices you'll have) people can give more specific brand and model suggestions.