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Does a big difference between P(Y) and P[Y | do(X=x)] imply that X has a strong causal effect on Y?
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$P(Y)$ and $P[Y | \text{do(X=x)}]$ represent the probability distribution of $Y$ and $Y$ given that there is an intervention on $X$. So the larger the difference between the outcome as a result of the intervention  and no action is an indicator of the impact of the intervention.

Yes it indicates a strong causal effect on $Y$.

Source: https://www.stat.cmu.edu/~larry/=sml/Causation.pdf
by Diamond (67,402 points)

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