Instructions on how to modify the Minecraft.app for OS X to work with Java 7 (without the need for 6). See the instructions in the "ReadMe" section.

ReadMe - Instructions.txt

These instructions are for Mac OS X 10.7.3 and later (those that can run Oracle's Java 7).

This assumes you have the Java 7 JDK installed from: http://www.oracle.com/technetwork/java/javase/downloads/index.html

When you download and install Minecraft.app into /Applications, you can control-click or right-click on the Minecraft.app file and "Show Package Contents".

Inside the "Contents" folder inside, there is a "Info.plist" file, which is plain text. Open this "Info.plist" file in a text editor of your choice and replace the contents with the "Info.plist" listed below.

Then open the Terminal and run the following commands:
touch /Applications/Minecraft.app/Contents/MacOS/LaunchGame.txt

This will create a blank text file called "LaunchGame" inside the Minecraft.app/Contents/MacOS folder.

Open the file and paste in the contents of the "LaunchGame" listed below.

Then run these commands in the Terminal:
mv /Applications/Minecraft.app/Contents/MacOS/LaunchGame.txt /Applications/Minecraft.app/Contents/MacOS/LaunchGame
chmod +x /Applications/Minecraft.app/Contents/MacOS/LaunchGame

As a last step - you should be able to double-click Minecraft.app and now it will launch using Java 7 (assuming it's installed). If you get an error message about an unknown developer, you may either have to adjust your Gatekeeper security settings in OS X or (easier) if you're an admin, you can just control-click / right-click on Minecraft.app and select "Open" and it will let you open the file successfully. Once you've done this once, double-click should work from that point on.

Info.plist

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist SYSTEM "file://localhost/System/Library/DTDs/PropertyList.dtd">
<plist version="1.0">
  <dict>
    <key>CFBundleDevelopmentRegion</key> <string>en</string>
    <key>CFBundleName</key> <string>MinecraftLauncher</string>
    <key>CFBundleVersion</key> <string>1.0.1</string>
    <key>CFBundleShortVersionString</key> <string>MinecraftLauncher 1.0.1</string>
    <key>CFBundleExecutable</key> <string>LaunchGame</string>
    <key>CFBundlePackageType</key> <string>APPL</string>
    <key>CFBundleSignature</key> <string>????</string>
    <key>CFBundleGetInfoString</key> <string>MinecraftLauncher 1.0.1 © Mojang Specifications, Inc, 2013</string>
    <key>CFBundleIconFile</key> <string>favicon.icns</string>
    <key>CFBundleAllowMixedLocalizations</key> <true/>
    <key>CFBundleInfoDictionaryVersion</key> <string>6.0</string>
    <key>CFBundleIdentifier</key> <string>com.Mojang Specifications.Minecraft.Minecraft</string>
    <key>LSHasLocalizedDisplayName</key> <true/>
    <key>CFBundleDisplayName</key> <string>Minecraft</string>
    <key>NSHighResolutionCapable</key> <true/>
  </dict>
</plist>

LaunchGame

#!/bin/bash
SCRIPTPATH=$(cd $(dirname $0);pwd -P)
/usr/bin/java -Xdock:icon="$SCRIPTPATH/../Resources/favicon.icns" -Xmx128m -jar "$SCRIPTPATH/../Resources/Java/Bootstrap.jar"

I'm having similar problems guys
Here's what my console says when I click on minecraft:
Last login: Wed Nov 16 10:45:02 on ttys000
FerdTrekampsMBP:~ ferdtrenkamp$ /Applications/Minecraft.app/Contents/MacOS/launcher ; exit;
2016-11-16 10:45:04.632 launcher[3581:146926] Claimed it found a path: /Applications/Minecraft.app/Contents/MacOS/launcher () 4096

Okay, finally got Minecraft working on my ancient Mac running OSX 10.7.5 (Lion, I believe, no I refuse to upgrade!).
The solution is to forget all of the above github, which did not work at all (missing bootstrap.jar, launcher exit, etc.), oh, except, do install the latest Java from Oracle (not Apple): http://www.oracle.com/technetwork/java/javase/downloads/index.html
What worked for me was to go to mc-launcher.com, click on Minecraft, click the Uncompressed MacOS jar file download. Click Skip Ad in the top right (the popups and main page are all spam!) and you will get TLauncher-MC.jar in Downloads. Double click it, open it, DO NOT install the java runtime for OSX from Apple if prompted, (install the one from Oracle), click Install and Enter, create an account by clicking the green plus sign (or add your existing mojang account), (I unchecked the elyby skins). Click Save Account. Click the Home icon in the bottom right. Then Install and Enter. That's it!

I'm using an early 2007 MacBook I've tried using this method on other apps and it bounces the app up once and then quits unexpectedly I think I know why correct me if im wrong but its a 64 bit app im trying to run and its a 86x64 system