Well, you certainly could, but you really shouldn't. Sending information in plain text in the URL is bad, since that URL may get cached in several different places and generally unnecessarily exposes sensitive information. You could encrypt it before sending it, but that's probably more trouble than it's worth.
The accepted method for using a purely URL based authentication is a token of some sort. You'll need to go through a standard login procedure to generate the token though. Addendum: Also note that that's better than putting the plain text password in the URL, but still has it's share of problems. URLs are more widely logged, cached and bookmarked than other header information. At least you retain control of the token though and can expire it after some time, whereas a password in the URL can't simply be expired.
If you're developing an API, I think it's reasonable enough to require some authentication procedure. If your clients can't do it, offer a library or code snippets that'll help them.