How Can I Only Allow Uniform Resizing in a WPF Window?

You can reserve aspect ratio of contents using WPF's ViewBox with control with fixed width and height inside.

Let's give this a try. You can change "Stretch" attribute of ViewBox to experience different results.

Here is my screeen shot:

<Window x:Class="TestWPF.Window1"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    Title="Window1" Height="300" Width="300">

    <Viewbox Stretch="Uniform">
        <StackPanel Background="Azure" Height="400" Width="300" Name="stackPanel1" VerticalAlignment="Top">
            <Button Name="testBtn" Width="200" Height="50">
                <TextBlock>Test</TextBlock>
            </Button>
        </StackPanel>
    </Viewbox>

</Window>

In the code sample:

if (sizeInfo.WidthChanged)     
{         
    this.Width = sizeInfo.NewSize.Height * aspectRatio;    
}     
else     
{         
    this.Height = sizeInfo.NewSize.Width * aspectRatio; 
} 

I believe the second computation should be:

this.Height = sizeInfo.NewSize.Width * (1/aspectRatio);  

I made a variation of this work in a "SizeChanged" event handler. Since I wanted the width to be the controlling dimension, I simply forced the height to match to it with a computation of the form:

if (aspectRatio > 0)
// enforce aspect ratio by restricting height to stay in sync with width.  
this.Height = this.ActualWidth * (1 / aspectRatio);

You may note the check for an aspectRatio > 0, ... I did this because I found that it was tending to call my handlers that did the resizing before the "Load" method had even assigned the aspectRatio.