How do I make my Rock-Paper-Scissor program more JavaScript-like?
Well, there might be a few things.

The most important thing that can be changed is removing all those magic numbers: What happens when we want to play “Rock, Paper, Scissors, Lizard, Spock” instead? You'd think you would only have to update the choiceArray – but unfortunately the length of that array is encoded in various places.

This means that if (userChoiceIndex &gt;= 0 &amp;&amp; userChoiceIndex &lt;= 2) should be 

if (userChoiceIndex &gt;= 0 &amp;&amp; userChoiceIndex &lt;= choiceArray.length - 1)


and var computerChoiceIndex = Math.floor(Math.random() * 3) should be

var computerChoiceIndex = Math.floor(Math.random() * choiceArray.length);


Your decideWinner function isn't really extensible, because you encode the winning relationships as code, not as the data it actually is. I'd use an object literal that only encodes what choice wins over another, like:

var winningData = {
  paper:     { rock:     true },
  rock:      { scissors: true },
  scissors:  { paper:    true }
};


Also, I'd implement decideWinner as a comparator that returns one of -1, 0, 1 depending on whether the first or the second argument won. This decouples the decision logic from UI like text.

function(choiceA, choiceB) {
  if (choiceA === choiceB)           return  0;
  if (winningData[choiceA][choiceB]) return -1;
  if (winningData[choiceB][choiceA]) return  1;
  console.warn("No ordering between " + choiceA + " and " + choiceB + " was found");
  return 0;
}


The game result is then determined as:

var ordering = decideWinner(userChoice, computerChoice);
var winningMessage = ordering &lt; 0 ? "You win!" :
                     ordering &gt; 0 ? "The computer wins!" :
                                    "It's a tie!";
var gameResult = "You chose: "          + userChoice     + ".\n" +
                 "The computer chose: " + computerChoice + ".\n" +
                 winningMessage;


Now when we extend the game, we just have to update the winningData. Notice that I display who wins, not what choice wins, as I don't think that would be tremendously interesting to a player. Also, he would have to make an association from the choice to the player, which is unnecessary mental effort.