I don't think that your site will be completely omitted from mobile search results. Here is the traffic data for one of my websites for a month.
My site doesn't have a mobile optimised version or a responsive design but still mobile and tablet constitute 12.4% of the visits.
Even though my site traffic isn't high to come to any conclusions, it answers your question :
Is true that Google will not show my site when a user searches on
their mobile phone.
The answer is No.
But I think that having a mobile optimised site will help you improve your rankings in mobile search. Otherwise, Google Webmasters Help would not be having an entire section devoted to Mobile Sites and wouldn't be using a different value for User Agent when crawling sites to index content for mobile searches. However, mobile optimisation is just one of the several ranking factors.