Sheikh Ahmad Fahad al-Sabah also vowed to release more information in the case against a former prime minister and the ex-speaker of parliament.
Public prosecutor Dherar al-Assoussi said Wednesday he would not press charges against ex-premier Sheikh Nasser Mohammad al-Ahmad al-Sabah and former parliament chief Jassem al-Khorafi.
The two officials were also accused of establishing contacts with foreign intelligence, money laundering and stealing public funds.
Sheikh Ahmad, a senior member of the ruling family and former energy minister, said in a statement released on Twitter that he will provide further evidence and "will not rest until those responsible are brought to justice".
In April last year, Sheikh Ahmad presented video recordings allegedly showing Sheikh Nasser and Khorafi planning a coup.
Assoussi said that examination by Kuwaiti security agencies showed that the recordings were not authentic and had been tampered with.
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Sheikh Ahmad has insisted the recordings were genuine, having secured a Swiss court ruling that the voices were those of the former two officials.
Sheikh Nasser, 75, resigned in 2011 after five years as premier following massive street protests over allegations that 13 lawmakers had received millions of dollars in bribes.
The two former officials, who were questioned by the prosecutor, dismissed the accusations as "fabrications and lies".
Khorafi, a wealthy businessman, was parliament speaker between 1999 and 2011. His lawyer said he will file a lawsuit for defamation against Sheikh Ahmad.
Sheikh Ahmad and Sheikh Nasser are both nephews of Kuwait's ruler, Emir Sheikh Sabah al-Ahmad Al-Sabah.
The case shook the oil-rich Gulf state, prompting the ruler to call for calm and to let the judiciary investigate.
Opposition groups had called for an international probe.