Dialog dialog = new Dialog(){Owner = System.Windows.Application.Current.MainWindow};
dialog.ShowDialog();

G>Dialog dialog = new Dialog(){Owner = System.Windows.Application.Current.MainWindow};
G>dialog.ShowDialog();
G>

Dispatcher.BeginInvoke((Action)(()=>
{
    Dialog dialog = new Dialog(){Owner = System.Windows.Application.Current.MainWindow};
    dialog.ShowDialog();
}));

public class ViewModel
{
    public Dispatcher Dispatcher { get; set; }
}

public ViewModel()
{
    Dispatcher = Dispatcher.CurrentDispatcher;
}

G>Пробовал через Dispatcher, Exception не выходит, но и окна тоже нет

<Window
    x:Class="ShowOwnedDialog.MainWindow"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    xmlns:l="clr-namespace:ShowOwnedDialog"
    Title="MainWindow" SizeToContent="Height" Width="500">
    
    <Window.DataContext>
        <l:MainWindowViewModel/>
    </Window.DataContext>

    <StackPanel Orientation="Vertical">
        <Button Content="В UI-потоке" Command="{Binding UIThreadCommand}"/>
        <Button Command="{Binding WorkThreadErrCommand}">
            <StackPanel TextBlock.TextAlignment="Center">
                <TextBlock Text="В рабочем потоке"/>
                <TextBlock Text="(с ошибкой the calling thread must be STA)"/>
            </StackPanel>
        </Button>
        <Button Command="{Binding WorkThreadNoErrCommand}">
            <StackPanel TextBlock.TextAlignment="Center">
                <TextBlock Text="В рабочем потоке"/>
                <TextBlock Text="(без ошибки - через Dispatcher.BeginInvoke)"/>
            </StackPanel>
        </Button>
    </StackPanel>

</Window>

class MainWindowViewModel
{
    public MainWindowViewModel()
    {
        Dispatcher = Dispatcher.CurrentDispatcher;

        UIThreadCommand = new RelayCommand(p =>
        {
            ShowDialogInternal();
        });

        WorkThreadErrCommand = new RelayCommand(p =>
        {
            new Thread(() =>
            {
                ShowDialogInternal();
            }).Start();
        });

        WorkThreadNoErrCommand = new RelayCommand(p =>
        {
            new Thread(() =>
            {
                Dispatcher.BeginInvoke((Action)(() =>
                {
                    ShowDialogInternal();
                }));
            }).Start();
        });
    }

    void ShowDialogInternal()
    {
        var dialog = new Dialog() { Owner = System.Windows.Application.Current.MainWindow };
        dialog.ShowDialog();
    }

    public Dispatcher Dispatcher { get; set; }

    public RelayCommand UIThreadCommand { get; private set; }

    public RelayCommand WorkThreadErrCommand { get; private set; }

    public RelayCommand WorkThreadNoErrCommand { get; private set; }
}