Can a C# class call an interface’s default interface method from its own implementation?

As far as I know you can’t invoke default interface method implementation in inheriting class (though there were proposals). But you can call it from inheriting interface:

public class HappyGreeter : IGreeter
{
    private interface IWorkAround : IGreeter
    {
        public void SayHello(string name)
        {
            (this as IGreeter).SayHello(name);
            System.Console.WriteLine("I hope you're doing great!!");
        }
    }

    private class WorkAround : IWorkAround {}

    public void SayHello(string name)
    {
        ((IWorkAround)new WorkAround()).SayHello(name);
    }
}

UPD

In my original answer too much wanted to show that you can call base in inheriting interface, but as @Alexei Levenkov suggested in comments cleaner way in this particular case would be something like this:

public class HappyGreeter : IGreeter
{
    private class WorkAround : IGreeter { }
    private static readonly IGreeter _workAround = new WorkAround();

    public void SayHello(string name)
    {
        _workAround.SayHello(name);
        System.Console.WriteLine("I hope you're doing great!!");
    }
}