Is Luck dependent on the Fortune Teller?

The fortune forecast just returns a phrase reflecting the value of the player's luck for that day. It doesn't change anything about the player's luck whether it's watched or not. The code tests the value of the player's luck and selects appropriate text to reflect it.

Also, the opening phrase (e.g. "Ah... I sense that a new viewer has joined us.") is purely random; it has no effect on gameplay.

(Source: Snooping through the code with a decompiler)

Although the wiki doesn't state much detailed information:

The Fortune Teller channel on the television will give an indication of the daily luck stat modifier.

A user in reddit did some code digging to learn more about luck modifiers and variants. As he/she states (emphasis mine):

Hey everyone, I wanted to clear up some information regarding luck. Right now, the official wiki article isn't correct about the meaning of each TV message, and the list of things luck affects is inaccurate and incomplete. I've done a lot of digging through the source code (1.06), and gathering info from other users, to understand how it works and what it does and doesn't affect. Hope you find this info useful! I tagged a lot of things as spoilers in case any new players want to discover them on their own.

Daily luck

It can range from -0.10 to +0.10, and is determined completely randomly when you go to sleep. The TV fortune forecaster will give show you one of six forecasts based on where your luck for the day falls within that range.

More information on https://www.reddit.com/r/StardewValley/comments/4c2gx4/a_comprehensive_guide_to_luck/, it's a very interesting thread btw.

So, long story short, luck is assigned when you go to sleep, the Fortune Teller only show you the random luck modifier you got.