It means "An irrevocable choice has been made" cowboy...

It means the person's fate or destiny is now established and cannot be changed. You could rewrite the sentence as: "...much the same inducements and alarms SEAL THE FATE of any tempted and trembling sinner." When people say, "The die is cast," they mean that a process has now begun and no one can change the result. They can no longer escape their fate. The idea behind the saying is that when you roll the dice, as soon as they leave your hand you cannot control the number you will get. Only your luck (or your fate/destiny) will determine the result.

Julius Caesar first said the words "the die is cast" (or rather he said it in Latin: "jacta alea est!") when he crossed the river Rubicon. This river was the boundary between ancient Italy and the province of Cisalpine Gaul. When Julius Caesar crossed the river in 49 B.C., he passed beyond the limits of his province of Cisalpine Gaul and became an invader in Italy, thus starting a war with Pompey and the Senate of Rome. In your extract the phrase is used as a metaphor to express the fact that there was no way back for the sinner..

Another equivalent idiom is "to cross the Rubicon" :)

it means to roll the dice, as in a game of dice. In other words, take a chance just as you would in a game of dice