Using a Grid is not the best way to achieve what you want. You should use a DockPanel instead with LastChildFill = "true". I can't try it now but I would imagine it like this:
<DockPanel LastChildFill="true">
<Expander Header=""
HorizontalAlignment="Left"
VerticalAlignment="Stretch"
ExpandDirection="Right"
IsExpanded="True"
DockPanel.Dock="Left"
Height="Auto"
>
<!-- My List control -->
</Expander>
<TabControl Name="ExecutionTab" HorizontalAlignment="Stretch">
<!-- soem tabs here -->
</TabControl>
</DockPanel>
This should make the tab control always take the entire remaining space.