Abstract property with public getter, define private setter in concrete class possible?

Unfortunately, you can’t do exactly what you want. You can do this with interfaces though:

public interface IInterface {
    string MyProperty { get; }
}

public class Class : IInterface {
    public string MyProperty { get; set; }
}

The way I would do it is to have a separate SetProperty method in the concrete classes:

public abstract class AbstractClass {
    public abstract string Value { get; }
}

public class ConcreteClass : AbstractClass {

    private string m_Value;
    public override string Value {
        get { return m_Value; }
    }

    public void SetValue(string value) {
        m_Value = value;
    }
}