Tevez was only appointed in June on a one-year contract but has decided to step down, citing politics within the club as his reason.
Rosario were supposed to hold their club elections last week but it has been postponed to December, with the club sitting in 20th position on the league table.
"I want to communicate to the Rosario Central fans that I'm going to step aside, I'm not going to continue," he said during a press conference.
"I want to thank the fans for these four and a half months, almost five months of support. The truth is that I have no words to thank everyone for the affection, the dedication and the sacrifice of all these months.
"[I'm taking this decision] always thinking of Rosario Central, the club always comes first. I'm stepping aside thinking about the club. I don't want to be an obstacle to anyone.
"Let whoever comes in have the freedom to make his own signings. The most sensible thing I can do is to step aside."
With regards to the club's presidential elections, the 38-year-old added: "From the first day I said I wasn't coming to do politics, and when my name is used [in politics], it's something I don't like. They [president candidates] weren't talking about a project, they were talking about Tevez.
"It's not a fair game for me, because I'm not from here, I'm not a Rosario Central born guy. I'm a manager and I don't want to do politics.
"[The elections] should have been last Sunday. Then it was postponed until December. It's something that catches us in the middle and you don't know who you're talking to. And if they postpone it until next year, we spent four or five months again with people who were not well disposed, angry. We've been managing it, with ups and downs, but we've been managing it.
"There is nothing else. As I have always said, Rosario Central comes first and not the people. When they want to put my name in politics, I won't allow that. I always want the best for Central."
Tevez retired in June with a record of 308 goals in 746 club games, having also turned out for the likes of Boca Juniors, Brazilian side Corinthians, West Ham United, Manchester United, Manchester City, Juventus and Shanghai Shenhua.
The 38-year-old won two Premier League titles and the UEFA Champions League with Manchester United before making a controversial switch to Manchester City, where he won another top flight title.
A move to Juventus in Italy brought two Serie A titles for Tevez.
Tevez won 76 caps for Argentina, scoring 13 goals and played at the 2006 and 2010 FIFA World Cups.