Dutch midfielder De Jong picked up two yellow cards in the second half and Koeman was dismissed in stoppage time for something he said to the fourth official.
The draw was Barca's second in LaLiga this week, following Monday's stalemate against Granda at Camp Nou, and further increases the pressure on Koeman.
The Dutch coach will now serve a touchline ban for Sunday's game against Levante, when another bad result could spell the end of his tenure at Barca, with sources telling ESPN earlier this week the club are already exploring replacements.
"I was sent off for telling the fourth official that there was a second ball on the pitch and the game should be stopped," Koeman said in a news conference after drawing with Cadiz.
"Everyone saw that except the referee. In this country, they send you off for nothing ...They say [it's for] 'attitude, attitude,' ... It is better we leave it there because it's not my problem."
Barca are now without a win in three games in all competitions with sources telling ESPN the relationship between coach and president Joan Laporta has deteriorated.
Laporta turned down the chance to explicitly back Koeman before Thursday's match and said that "if decisions need to be made, we will make them" following Barca's slow start to the season.
"It's not just about the result, you have to analyse the attitude and the work rate," Koeman, who refused to take questions from the media on Wednesday, said when asked if he feared for his future after Barca's third draw in five league outings this season.
"I can't complain. We've had four or five chances and we haven't scored, so for that reason I am not happy. But what I will take with me is the attitude of the team and not my personal situation."
"People take a title and change the context. I also said we would compete this season. I'm not here wearing the Barca shirt to come second or third. I am here to compete for every trophy.
"And I am convinced, despite the start we've made, that we will compete. The team's motivated. Sometimes you have to go through these moments.
"We know the fans are with us despite the situation, which is complicated for everyone. It's been 12 years at the top. Now we all have to stick together. We will compete until the end, don't doubt that.
"This club has been riding the crest of a wave for many years and maybe we're not used to this. It's been a convulsive period, a change of president, coaches... we have to do all we can to provide stability.
"We all want to win. Let's not look for sides, we can all complain or we can all pull in the same direction. The players are here to pull in the same direction. We are focused on playing and winning."
Barca return to LaLiga action on Sunday against Levante at Camp Nou, with Ansu Fati in line to make his return following 10 months out with a knee injury. They then travel to Benfica and Atletico Madrid before the October international break.