I gave up trying to get my iPhone to play nicely with CSS, and had to resort to using jQuery.
In my webpage, I added a <div> which I want to fill the screen:
<body>
    <div class="cssFullScreen" />
    ...etc...
Then I added in two tablespoons of CSS...
<style>
    .cssFullScreen
    {
        position: absolute;
        left:0px;
        top:0px;
        background: url(BackgroundImage.jpg) no-repeat center center fixed; 
        -webkit-background-size: cover;
        -moz-background-size: cover;
        -o-background-size: cover;
        background-size: cover;
    }
</style>
..and a reluctant scoop of jQuery...
<script src="jquery-2.2.3.min.js"></script>
<script type="text/javascript">
    $().ready(function () {
        ResizeWindows();
        $(window).resize(function () {
            ResizeWindows();
        });
    });
    function ResizeWindows() {
        //  When the user resizes the window, we'll resize any DOM elements
        //  with the "cssFullScreen" class.
        var width = window.innerWidth ? window.innerWidth : $(window).width();
        var height = window.innerHeight ? window.innerHeight : $(window).height();
        $(".cssFullScreen").width(width);
        $(".cssFullScreen").height(height);
    }
</script>
It's not pretty, but it was the only thing I could find which really worked on an iPhone.
And strangely, this code only worked (on an iPhone) when applied to a div.  If I tried to apply it directly to the html or body, it did nothing...