Granger proposed a test for causality based on time series, is there any relation with the notion of causality by Rubin that seems prevalent on applied micro work. The latter deals with the notion of counterfactual and how we can use an experimental set-up to capture the expected value of this parallel world situation to infer about the average effect of the intervention.
In short, not really. Prof. John Cochrane offers an astute example of Granger causality by stating that a weatherman Granger-causes the weather. We know that a weatherman has no measurable role in causing the weather, but there is indeed an intertemporal correlation between what the weatherman says today and what happens tomorrow.
Sir Clive Granger was a master not only of Time-Series Econometrics, but also of giving very appealing names to his works. With his concept of "causality" I guess he went a bit too far: reading the original paper, p.428,
...and "better able to predict" is essentially defined in the first sentence (reduced prediction error variance, which is $\sigma^2 (X\mid U)$, conditional on $U$, that includes $Y$).
In other words, Granger shamelessly used the word "causality" as a synonym for "predictive power": if information on $Y$ improves our predictions on $X$ then "$Y$ is causing $X$". His implicit argument is clear: if they correlate there may be also a causative link here -who knows?... No wonder scholars felt the need to subsequently name it "Granger-causality" -proving Granger's strategic mastery in name-giving!
So no, no relation to Rubin's concept.