What is the heaviest grain bullet that I can shoot out of a 20" 1 in 9 twist H-barrel safely and accurately (stable).

                                            Bri

Denpends on the barrel and the round - rounds are stabilized based on length not weight.  (though length and weight are usually related). Mine will shoot to 77gr, others listers have reported theirs won't do well over 73gr, and still others 1:9 twist rifles will only shoot up to 69gr well.  There is too much variation in the manufacturing process to say how well YOUR barrel will shoot the heavier bullets.