This is regretfully a behavior of all Classic elements inside Cornerstone. I'd recommend that you use the Social element instead. Not only that it will display inline in Cornerstone and front-end, it also has a link option.
If you wish to add text to each individual icons, I'd recommend you use the Button element which is also an inline element.
In case you don't need linking the icons, use the Icon shortcode instead.
Hope that helps.