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SHORT ANSWER
General Gursel wrote in a letter that democracy could not be built on an exceution and he promised, in the aftermath of the coup, that there would be no "mistreatment" of Adnan Menderes. Also, he knew the execution would not go down well on the international stage. Further, as a conciliatory figure, he was eager to bring his country together; executing Menderes would undermine national unity.
DETAILS
The 1960 coup was primarily the work of younger, middle-ranking officers who were pressured into chosing General Gursel to head the National Unity Committe (NUC). Gursel, who at the time was retired but still popular both within and outside the military, was a democrat and, as head of state, immediately set about forming a new, more liberal constitution and a return to democracy.
Cemal Gürsel resisted attempts to continue military rule. He was elected as the fourth President of the Republic in 1961 and played an important role in the preparation of a new Constitution and return back to the democracy after the coup.
Later, in an open letter, Gursel wrote:
İdam üzerinden demokrasinin kurulamayacağını bu toplum mutlak bir gün anlayacaktır
Google translates this (somewhat clumsily) as:
This society will understand an absolute day that democracy cannot be built up through the execution
Gursel adds:
Büyük bir haksızlık yapıldı kanısını hala taşımaktayım.
which Google translates as:
I still carry my belief that a great injustice was done.
The general ends his letter, poignantly, with
Üzgünüm Menderes.
Sorry Menderes.
On the international stage, maintaining good relations with the United States was seen as essential to the country’s long-term security. Ömer Aslan, in The United States and Military Coups in Turkey and Pakistan, notes that
The US National Security Council, which convened on June 30, 1960, observed after the 1960 coup that ‘the regime in power is not as pro-Western as the Menderes regime was'.
However, the United States appreciated the fact the May 27 coup had been bloodless; that would no longer be the case in members of the previous government were executed.
Gursel’s concern about the international reaction to the decision of the NUC appointed tribunal to execute Menderes and two others quickly proved to be valid. The newspaper Dawn on Sept 17, 1961 reported
The Turkish government was yesterday subjected to considerable pressure from both foreign and domestic sources to grant a reprieve to the former premier. The British Ambassador, Sir Bernard Burrows, was received by President Gen Cemal Gursel and French Ambassador Henri Spitzmuller had seen Foreign Minister Selim Sarper.
The paper adds,
These interventions were viewed with disfavour by the young officers’ group which has not concealed its disappointment that 12 of the 15 persons sentenced to death in the mass treason trial have already been reprieved.
Why didn’t Gursel intervene to stop the execution? According to Wikipedia, he tried.
The revolution tribunal's chief prosecutor Egesel conducted the execution despite not being authorized. İsmet İnönü and Cemal Gürsel were already phoning for him (Menderes) not to be executed but the telecommunications' office cut off the lines and Egesel made use of the (communication) gap to conduct the execution.
It is also worth noting that Gursel did not have the universal support of the military when he pushed for a return to democracy, as proved by a military assassination attempt.
Other Sources:
Erik J. Zurcher, Turkey: A Modern History
Coup D'Etat: Untold Story of Washington Leaving Its Friends in the Lurch (2016)