"Today searches were carried out to my house, to my company and to the Jose Alvalade Stadium, as part of a process in which I was charged with defamation," Cristovao said in the statement reported by state news agency LUSA.
"My decision to quit does not mean that I accept that I am guilty, on the contrary, it is only to serve the higher interests of Sporting," he added.
Local media reported that Cristovao had taken steps to damage the reputation of match referee Jose Cardinal ahead of the Portuguese Cup quarter-final between Sporting and Maritimo at the Alvalade on December 22.
The prosecutor's office said it had opened an investigation into the affair in December after Portuguese Football Federation president Fernando Gomes had raised concerns about illegal activity.
The FPF took Cardinal off the Cup match for what it called "personal reasons". Sporting beat fellow top flight side Maritimo 3-0 and have since reached the final in which they will play Academica on May 20.
Sporting issued a statement on their official website on Thursday saying: "Searches were made at the Jose Alvalade stadium by the prosecutor's office and police and we fully cooperated."
Sporting added that Cristovao, a former police inspector, had asked to be relieved of his duties but the club said they could not provide further details about the reasons for his request or why the police had searched the club.
Referee Cardinal, 45, a former postman from Porto, was a linesman at the 2010 World Cup finals in South Africa.