If you are using a raster image (jpg/png/etc), the first idea that comes to mind would be to use the standard Picture Fill option, then adjust the map view so you can see your entire polygon. Next, increase the Size setting on the picture fill until your image covers the entire polygon without any repeats or seams. It may need to be a very large number, depending on the image size, and you might need a very large (high resolution) image. Once you find the right size, go to Contents pane and right click your Map item, then choose Set Reference Scale. That will stop the image from resizing on the fly.
SVG would be easier, I think. Use the Marker Fill option instead, and choose your SVG file. Under the Marker Placement dropdown, set the Placement option to "Center". This will put just one copy of the svg file in the middle of the polygon. Then adjust the polygon size until it's filling the polygon as much as you want. You do have the option to connect the Size property to an attribute, so if you have a bunch of different size polygons you could also use a Shape Area field or similar and divide/multiply the value to have the sizing done for each polygon on the fly. You might also want to check the option to "Clip marker to boundary" depending on the look you are going for.
-Aaron
