What is the song that is playing over the closing credits of the movie
"When a Man Loves a Woman", 1994?  It is not on the soundtrack.

susan469 --

The song you are looking for is almost certainly "Running From Mercy,"
as sung by Rickie Lee Jones.

The evidence on which my conclusion is based is as follows:

1.  The Internet Movie Database lists the five non-original songs that
are performed in the 1994 film, "When a Man Loves a Woman."  One of
them is "Running From Mercy" by Rickie Lee Jones.  Here is a link to
that list:
IMDb: Soundtracks for "When a Man Loves a Woman"
http://us.imdb.com/Soundtracks?0111693

2.  The CD soundtrack to the film does not include three of those
songs, including "Running From Mercy."  Here is a link to the contents
of that CD:
Amazon.com: "When a Man Loves a Woman" (Soundtrack)
http://www.amazon.com/exec/obidos/tg/stores/detail/-/music/B000000ODC/glance/ref=cm_cr_dp_2_1/002-6126080-7143267

3.  Many of the "customer reviews" of the soundtrack at Amazon.com
complain about the fact that some of the music in the film is not on
the CD.  Two of those reviewers mention that the song used in the
closing credits (which is among those not on the soundtrack CD) is
sung by Rickie Lee Jones.  The first such reviewer says that the song
is "Stewart's Coat".  However, a subsequent reviewer has this to say:
"By the way, the song in the credits was indeed by Rickie Lee Jones,
but the title was 'Running From Mercy' and not 'Stewart's Coat' as
previously posted by someone else."  This comment is in a review
entitled "Great Movie - Lousy Soundtrack" (February 8, 2002), about
halfway down the following linked page:
Amazon.com: "When a Man Loves a Woman": Customer Reviews
http://www.amazon.com/exec/obidos/tg/detail/-/B000000ODC/ref=cm_cr_dp_2_1/002-6126080-7143267?v=glance&s=music&vi=customer-reviews

4.  Thus, the only two reviewers mentioning the film's closing credits
agreed that the closing credits included a Rickie Lee Jones song. 
This, and because the IMDb song listing identifed "Running From Mercy"
as the only Rickie Lee Jones contribution to the soundtrack, leads me
to believe that this is the song you are seeking.  Here is a link to a
page containing a brief sample of that song, as contained on the
Rickie Lee Jones CD entitled "Traffic From Paradise":
Tower Records: "Traffic From Paradise"
http://www.towerrecords.com/product.asp?pfid=1034449&cc=USD


Search Strategy:

Searched the Internet Movie Database for the contents of the film's
soundtrack.
Searched various CD retailers for information on the soundtrack CD.
Searched the All Music Guide ( http://www.allmusic.com )for a Rickie
Lee Jones CD containing "Running From Mercy."


I am confident that this is the information you are seeking.  If any
of the above needs clarification, or if you are dissatisfied with the
answer for any reason, please ask for clarification before rating the
answer.

markj-ga

It was extremely helpful that you were so descriptive about where you
got your information from.  Although the exact answer you gave was not
the one I was looking for, your explanation led me to the correct
answer.