Duggar, 27, is the oldest of the children who appear on TLC's hit show "19 Kids and Counting." The Duggars are known for being devout Christians who don't believe in practicing birth control and whose children follow strict courtship rules.
TLC pulled all episodes of the show currently set to air, according to Shannon Llanes, a spokeswoman for the network.
The network had already replaced
several scheduled repeats of "19 Kids and Counting" with "The Little People."
"Twelve years ago, as a young teenager I acted inexcusably for which I am extremely sorry and deeply regret. I hurt others, including my family and close friends," Josh Duggar said in a post on Facebook
"I confessed this to my parents who took several steps to help me address the situation. We spoke with the authorities where I confessed my wrongdoing and my parents arranged for me and those affected by my actions to receive counseling."
He also resigned from his position at the Family Research Council, a nongovernmental organization that says its mission
is to "advance faith, family and freedom in public policy and the culture from a Christian worldview."
A statement from the national faith-based group said he did so "as a result of previously unknown information becoming public concerning events that occurred during his teenage years."
The Duggar family has used its celebrity to back socially conservative causes and candidates, which could pose a problem for Republican presidential hopefuls
who have aligned themselves with the Duggars.
Duggar's statement appears to refer to a 2006 police report obtained by In Touch Weekly detailing the accusations and investigation of child molestation in Springdale, Arkansas. The report is heavily redacted and does not include the name of the person accused of "forcible fondling," but the dates listed in the document seem to line up with Duggar's statement Thursday.
On Thursday, a judge ordered that police report obtained by In Touch Weekly be expunged because one of the victims -- a minor -- could be "directly or indirectly" identified in it.
Lt. Scott Lewis of Springdale police told CNN that the police report cited by In Touch Weekly had been expunged because of a judge's order.
Josh Duggar's parents, Jim Bob and Michelle Duggar, who are named in the police report, also spoke out Thursday.
"Back 12 years ago our family went through one of the most difficult times of our lives. When Josh was a young teenager, he made some very bad mistakes and we were shocked. We had tried to teach him right from wrong," they said on Facebook.
"That dark and difficult time caused us to seek God like never before. Even though we would never choose to go through something so terrible, each one of our family members drew closer to God."
The "19 Kids and Counting" series began in 2008 when the Duggar children numbered 17. It's one of TLC's top-rated programs, and the show was heavily featured during a presentation to advertisers last month.
whether or not the show will go forward. Last season's premier of "19 Kids and Counting" was the highest-rated debut yet, according to TLC.
But some note that the network cancelled "Here Comes Honey Boo Boo"
after a tabloid linked a character on the show to a convicted child molester.
TLC declined to comment on the Duggars' statements Thursday.