The decision drew an angry response from Cardiff City, who play in League One, the third tier of competition, who said in a statement: "The world of football must take a hard look at itself."
The Welsh club were seeking damages of more than $192 million (£100 million), covering Sala's transfer fee but also losses of other potential earnings.
Sala died in January 2019 when the light aircraft he was travelling in from France to link up with Cardiff crashed into the English Channel. Pilot David Ibbotson, 59, also died in the crash.
Cardiff's contention was that Nantes were responsible because the flight was organised by an agent, Willie McKay, enlisted by the French club.
However, the tribunal of Nantes dismissed the claim on Monday, and ordered Cardiff to pay more than $770,000 to the French club.
Cardiff issued a statement which read: "We deeply regret that the court did not recognise FC Nantes' liability in this tragedy.
"We initiated these proceedings so that the full truth of this case could come to light, in respect of Emiliano Sala's memory.
"Today, we note with bitterness that the principles of transparency, integrity and safety in professional football have not prevailed in this decision — as illustrated by the impunity enjoyed by Willie McKay, a controversial and unlicensed agent, a situation that did not appear to raise concern for FC Nantes, and whose actions contributed to the death of two men.
"How is it possible that Willie McKay can still operate after such a tragedy?
"How is it possible that, in the end, FC Nantes — which never contributed to the support fund we launched in memory of Emiliano — paid more money to Willie McKay ($2.5 million) than to the Sala family?
"More than ever, there is an urgent need to strengthen safety and ethical standards in the management and protection of players, so that such a tragedy can never happen again.
"The world of football must take a hard look at itself."
In a separate process, Cardiff were ordered by football's global governing body, FIFA, to pay Nantes the three instalments of the transfer fee agreed for Sala before the Argentinian's death, totalling $25 million.
The Court of Arbitration for Sport upheld Nantes' claim related to the payment of the first instalment in August 2022, while Cardiff announced in June 2023 that they had been ordered to pay the remaining two instalments by FIFA as well.
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