It's a nice idea but there are a couple of things to bear in mind.
Firstly, we have to be very careful about copyright, and much depends on which version is being quoted. If using the NIV for example, it's important to remember that the whole thing is copyright protected. Provision is made for quoting it as follows:
The NIV text may be quoted in any form (written, visual, electronic or
audio), up to and inclusive of five hundred (500) verses without
express written permission of the publisher, providing the verses do
not amount to a complete book of the Bible nor do the verses quoted
account for twenty-five percent (25%) or more of the total text of the
work in which they are quoted.
When the NIV is quoted in works that exercise the above fair use
clause, notice of copyright must appear on the title or copyright page
or opening screen of the work (whichever is appropriate) as follows:
THE HOLY BIBLE, NEW INTERNATIONAL VERSION®, NIV® Copyright © 1973,
1978, 1984, 2011 by Biblica, Inc.™ Used by permission. All rights
These Scriptures are copyrighted by the Biblica, Inc.™ and have been made available on the Internet for your personal use only. Any other use including, but not limited to, copying or reposting on the Internet is prohibited.
The last paragraph I believe applies explicitly to copying from The Bible Gateway.
While we're normally only quoting one or two verses at a time in a post, this could soon add up to more than 500 verses or a complete book across the SE site, and I believe that could put StackExchange on unsafe legal ground. We would also need to get the copyright notice onto the relevant SE pages (and decided which pages they are) which could itself be problematic.
Secondly, different people have different preferred versions and translations of the Bible. If the purpose of referencing a piece of scripture is to get at the truth and meaning behind it, as opposed to the exact wording used in one particular version, then I would argue it's better just to supply a reference so that users are at liberty to choose which version they want to use.
In circumstances where you do want to use a particular version I would advocate linking to the passage on BibleGateway.com rather than quoting it directly so that we don't have to worry about that copyright problem.