eclipse doesn't start:

Have you tried one or more of:
1. double clicking /Applications/eclipse/Eclipse.app in Finder
2. open /Applications/eclipse/Eclipse.app
3. /Applications/eclipse/Eclipse.app/Contents/MacOS/eclipse

?

More questions:

Did you download eclipse 3.3.1.1 onto your iMac or did you use the same tgz that installed properly on the MacBook Pro? If it was a new download, you might check that it downloaded correctly, for example, by checking the md5 hash ( md5 filename in Terminal). Compare to the one at http://download.eclipse.org/eclipse/downloads/drops/R-3.3.1.1-200710231652/index .php For example, for
eclipse-SDK-3.3.1.1-macosx-carbon.tar.gz (is that what you installed?) the MD5 should be 66deca5411648105134db084d4a95aa9.

Are you using the same user to try to run Eclipse as installed it? If stuffit didn't set permissions correctly, that might be an issue. You could try
sudo chmod go+Xr /Applications/eclipse
just to be sure.

Finally, looking at your stack trace more closely again, I realize that what may be happening is that one or more of the jar files which contain the eclipse code is "sealed" securely. The Exception you are receiving may be saying that Java is unable to confirm the authenticity of the code, and so Java is refusing to run it. Unfortunately, Java is also refusing to tell us which jar file is the problem. Hopefully this is just a download issue, but you might need to install key certificates or something horrible like that. You might try editing /Applications/eclipse/eclipse.ini by appending one or more of the lines
-verbose
-verbose:class
-verbose:jni
which, if I'm not mistaken, will hopefully give more diagnostic information when you try /Applications/eclipse/eclipse. Other possibly useful flags (one per line) which I'm less optimistic about being helpful here are -ea and -esa to enable assertions and system assertions. None of these will solve the problem, but one or more might make it more obvious what the problem is.

I hope the problem isn't missing system key certificates. I would try confirming the original tar.gz. Unpack it, and copy it into /Applications, overwriting the /Applications/eclipse which is already there (or delete that one first). Run it by double clicking /Applications/eclipse/Eclipse.app, and not by using a Terminal command. Cross your fingers...

First, I didn't notice any Java in the link you quoted. Perhaps I'm blind. But before you try to reinstall Java, please humour me with the following. Go to, say, http://jeremy.smallinfinity.net/games/react/ Does the applet work? (You said the dancing Duke works; I chose this link because there is a standalone version of it that we are getting to in a minute.) If it doesn't work, then I'm satisfied that there is something wrong with your Java. If it does work, then download the "application" version and launch it. If that reproduces the error, then again I'm satisfied that there is something wrong with your Java. If not, then I have to find a signed Java app to test... 🙂

Next I should warn you that I failed to help someone else reinstall Java in another thread. So before I give you similar instructions that are surely doomed to failure, let me ask: How did you install Leopard? Erase and install? Archive and install? Upgrade? (Is the last even an option?) If you did an upgrade (or even an archive and install) you might consider doing (another) archive and install. That may fix things, and is probably the easiest way to "reinstall" Java. Hopefully it won't break anything...I'm willing to assume that an erase and install should have worked, so we should proceed...

I notice that the Leopard install DVD contains the files System/Installation/Packages/Java.pkg and System/Installation/Packages/JavaTools.pkg. System is a hidden directory on the DVD, so you will either have to access them via Finder and the "Go > Go To Folder" menu item (or its French equivalent), via Terminal, or by making hidden files visible in Finder through
defaults write com.apple.finder AppleShowAllFiles -bool true; killall Finder
I don't know in which order you should install them. I suspect that if you try to install them, the installation will fail with the message that they are already installed. To eliminate that message and force the installation, delete all files in /Library/Receipts that appear to be Java related. I seem to have
/Library/Receipts/boms/com.apple.pkg.Java.bom
/Library/Receipts/boms/com.apple.pkg.JavaSDKLeo.bom
/Library/Receipts/boms/com.apple.pkg.JavaToolsLeo.bom
Your results may vary. Then double-clicking the two pkg's (if you could access them in Finder) or open /Volumes/Mac\ Os\ X\ Install\ DVD/System/Installation/Packages/Java.pkg and similarly for JavaTools.pkg (not necessarily in that order) will hopefully install Java for you. If we are lucky, it will even do so correctly.

Backup first! Bon chance! Feel free to ignore these suggestions!

Cool applet!

One more idea from me; sorry I'm striking out.

<pre style="line-height:50%">
$ ls -al /System/Library/Frameworks/JavaVM.framework/Versions/1.5.0/Classes/
total 56784
drwxr-xr-x 11 root wheel 374 Nov 6 02:14 .
drwxr-xr-x 8 root wheel 272 Nov 15 10:00 ..
drwxr-xr-x 3 root wheel 102 Nov 6 02:14 .compatibility
-rw-r--r-- 1 root wheel 4630797 Nov 6 01:58 charsets.jar
-rw-r--r-- 1 root wheel 22528437 Nov 6 02:05 classes.jar
-rw-r--r-- 1 root wheel 136636 Nov 6 02:05 dt.jar
-rw-r--r-- 1 root wheel 82642 Nov 6 01:45 jce.jar
-rw-r--r-- 1 root wheel 232629 Nov 6 01:59 jconsole.jar
-rw-r--r-- 1 root wheel 278473 Nov 6 01:45 jsse.jar
-rw-r--r-- 1 root wheel 852316 Nov 6 02:14 laf.jar
-rw-r--r-- 1 root wheel 312273 Nov 6 02:07 ui.jar
</pre>
In particular, it is classes.jar which contains the sun.security.util.SignatureFileVerifier.processImpl method which is throwing the exception. To be more precise
<pre style="line-height:50%">
$ jar -tvf /System/Library/Frameworks/JavaVM.framework/Versions/1.5.0/Classes/classes.jar | egrep '(SignatureFileVerifier.class|JarVerifier.class|JarFile.class|URLClassPath.clas s|/Resource.class|java/net/URLClassLoader.class|java/security/AccessController.c lass)'
10817 Mon Nov 05 22:48:00 EST 2007 sun/security/util/SignatureFileVerifier.class
4274 Mon Nov 05 22:47:52 EST 2007 sun/net/www/protocol/jar/URLJarFile.class
1625 Mon Nov 05 22:48:00 EST 2007 sun/misc/Resource.class
6208 Mon Nov 05 22:48:00 EST 2007 java/util/jar/JarVerifier.class
6637 Mon Nov 05 22:48:00 EST 2007 java/util/jar/JarFile.class
6252 Mon Nov 05 22:47:58 EST 2007 sun/misc/URLClassPath.class
8285 Mon Nov 05 22:48:00 EST 2007 java/net/URLClassLoader.class
2200 Mon Nov 05 22:48:00 EST 2007 java/security/AccessController.class
</pre>
Do your jar files match mine and/or those of your MBP on which eclipse works? If not assuming (but when you assume...) that the jar files contain only java byte code, not native code, I suspect you could copy the jar files from your MBP to your G5 and see if that solves the problem.

Have a look at http://discussions.apple.com/thread.jspa?threadID=1218367&tstart=0
There are a few suggestions there that seem reasonable, like manually installing the 10.5.1 combo update instead of that from software updates. Also, comments about (not) running 3rd party apps.