Wow- very interesting comment from Google’s John Mueller (Sr. Webmaster Trends Analyst) this morning around HTTPS that flies in the face of everything I’ve heard over the past year. Responding to someone on Twitter who stated that “Without an HTTPS certificate, Google is likely to drop your website in search results.” John replied "Yeah, that’s wrong." “HTTPS is not a factor in deciding whether or not to index a page, at all,” he added.
He did explain that Google does “use HTTPS as a light-weight ranking factor, and having HTTPS is great for users.”
That said, all of my sites are HTTPS - and I certainly wouldn’t put up any new ones that aren’t…but, he says loud and clear that your sites do not need to be HTTPS to be indexed and ranked by Google.
I’ve never heard or seen it said anywhere that not having https will get a site dropped. Only that it may, MAY, impact results.
I agree with @steveb, never heard anyone (until this post) suggest that Google or any search engine was dropping Websites from SERP listings based on HTTPS.
Now Chrome will warn users a site is non-secure if it’s not HTTPS. That will definitely hurt your conversion rate.
For my regular 9-5 job, I create and maintain websites for a huge integrated health system. Our marketing department receives two or three unsolicited offers each week to improve our SERP (which is already top-notch). Without fail, the SEO “experts” mention https in their sales pitch (which is rather humorous since all of our sites have long been secured). And, most, if not all, of the legitimate SEO companies also mention it- although they do say that as of now it’s a minor weight factor.
As @steveb and @Teefers say; Google will add more weighting to secure websites, but should not totally remove insecure websites from their searches.
There are still enormous numbers of websites out there that do not have SSL, and possibly never will have it.
If a website is identified as having a major security flaw, then it too will still get shown in search results. But you will likely see a big ‘skull and crossbones’ warning error, if you attempt to go to it.