Maintain aspect ratio of div but fill screen width and height in CSS?

Use the new CSS viewport units vw and vh (viewport width / viewport height)

FIDDLE

Resize vertically and horizontally and you’ll see that the element will always fill the maximum viewport size without breaking the ratio and without scrollbars!

(PURE) CSS

div
{
    width: 100vw; 
    height: 56.25vw; /* height:width ratio = 9/16 = .5625  */
    background: pink;
    max-height: 100vh;
    max-width: 177.78vh; /* 16/9 = 1.778 */
    margin: auto;
    position: absolute;
    top:0;bottom:0; /* vertical center */
    left:0;right:0; /* horizontal center */
}

* {
  margin: 0;
  padding: 0;
}
div {
  width: 100vw;
  height: 56.25vw;
  /* 100/56.25 = 1.778 */
  background: pink;
  max-height: 100vh;
  max-width: 177.78vh;
  /* 16/9 = 1.778 */
  margin: auto;
  position: absolute;
  top: 0;
  bottom: 0;
  /* vertical center */
  left: 0;
  right: 0;
  /* horizontal center */
}

<div></div>

If you want to use a maximum of say 90% width and height of the viewport: FIDDLE

* {
  margin: 0;
  padding: 0;
}
div {
  width: 90vw;
  /* 90% of viewport vidth */
  height: 50.625vw;
  /* ratio = 9/16 * 90 = 50.625 */
  background: pink;
  max-height: 90vh;
  max-width: 160vh;
  /* 16/9 * 90 = 160 */
  margin: auto;
  position: absolute;
  top: 0;
  bottom: 0;
  left: 0;
  right: 0;
}

<div></div>

Also, browser support is pretty good too: IE9+, FF, Chrome, Safari- caniuse