This is a patch to support the docker version 1.6.0 or greater for EB CLI 3.7.3.

compat.py.patch

--- compat.py.bk	2016-03-10 07:43:01.000000000 +0900
+++ compat.py	2016-03-10 07:45:42.000000000 +0900
@@ -29,7 +29,9 @@ def supported_docker_installed():
     """
 
     try:
-        return commands.version() >= SUPPORTED_DOCKER_V
+        # return commands.version() >= SUPPORTED_DOCKER_V
+        from distutils.version import LooseVersion
+        return LooseVersion(commands.version()) >= LooseVersion(SUPPORTED_DOCKER_V)
     # OSError = Not installed
     # CommandError = docker versions less than 1.5 give exit code 1
     # with 'docker --version'.

$ cd /usr/local/Cellar/aws-elasticbeanstalk/3.7.3/libexec/lib/python2.7/site-packages/ebcli/containers
$ curl -LO https://gist.githubusercontent.com/nori3tsu/015c33e2db5bc823c2de/raw/3e781b7c22c492bc8bc77140b90b3590c0770950/compat.py.patch
$ patch -u compat.py < compat.py.patch

The AWS forum thread for this is at:

https://forums.aws.amazon.com/message.jspa?messageID=701953

fwiw the path on my machine was

/Library/Frameworks/Python.framework/Versions/3.5/lib/python3.5/site-packages/ebcli/containers/compat.py