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

Just reformulating Danield's answer in a LESS mixin, for further usage:

// Mixin for ratio dimensions    
.viewportRatio(@x, @y) {
  width: 100vw;
  height: @y * 100vw / @x;
  max-width: @x / @y * 100vh;
  max-height: 100vh;
}

div {

  // Force a ratio of 5:1 for all <div>
  .viewportRatio(5, 1);

  background-color: blue;
  margin: 0;
  position: absolute;
  top: 0; right: 0; bottom: 0; left: 0;
}

Use a CSS media query @media together with the CSS viewport units vw and vh to adapt to the aspect ratio of the viewport. This has the benefit of updating other properties, like font-size, too.

This fiddle demonstrates the fixed aspect ratio for the div, and the text matching its scale exactly.

Initial size is based on full width:

div {
  width: 100vw; 
  height: calc(100vw * 9 / 16);

  font-size: 10vw;

  /* align center */
  margin: auto;
  position: absolute;
  top: 0px; right: 0px; bottom: 0px; left: 0px;

  /* visual indicators */
  background:
    url(data:image/gif;base64,R0lGODlhAgAUAIABAB1ziv///yH5BAEAAAEALAAAAAACABQAAAIHhI+pa+EPCwA7) repeat-y center top,
    url(data:image/gif;base64,R0lGODlhFAACAIABAIodZP///yH5BAEAAAEALAAAAAAUAAIAAAIIhI8Zu+nIVgEAOw==) repeat-x left center,
    silver;
}

Then, when the viewport is wider than the desired aspect ratio, switch to height as base measure:

/* 100 * 16/9 = 177.778 */
@media (min-width: 177.778vh) {
  div {
    height: 100vh;
    width: calc(100vh * 16 / 9);

    font-size: calc(10vh * 16 / 9);
  }
}

This is very similar in vein to the max-width and max-height approach by @Danield, just more flexible.

My original question for this was how to both have an element of a fixed aspect, but to fit that within a specified container exactly, which makes it a little fiddly. If you simply want an individual element to maintain its aspect ratio it is a lot easier.

The best method I've come across is by giving an element zero height and then using percentage padding-bottom to give it height. Percentage padding is always proportional to the width of an element, and not its height, even if its top or bottom padding.

W3C Padding Properties

So utilising that you can give an element a percentage width to sit within a container, and then padding to specify the aspect ratio, or in other terms, the relationship between its width and height.

.object {
    width: 80%; /* whatever width here, can be fixed no of pixels etc. */
    height: 0px;
    padding-bottom: 56.25%;
}
.object .content {
    position: absolute;
    top: 0px;
    left: 0px;
    height: 100%;
    width: 100%;
    box-sizing: border-box;
    -moz-box-sizing: border-box;
    padding: 40px;
}

So in the above example the object takes 80% of the container width, and then its height is 56.25% of that value. If it's width was 160px then the bottom padding, and thus the height would be 90px - a 16:9 aspect.

The slight problem here, which may not be an issue for you, is that there is no natural space inside your new object. If you need to put some text in for example and that text needs to take it's own padding values you need to add a container inside and specify the properties in there.

Also vw and vh units aren't supported on some older browsers, so the accepted answer to my question might not be possible for you and you might have to use something more lo-fi.