The Korean War and the Vietnam War are fundamentally very different conflicts.
With the former, South Korea was unambiguously invaded by North Korea in an explicit war of conquest. Due to lucky political circumstances, United Nations was able to sanction a military operation to defend South Korea.
The Security Council,
Having determined that the armed attack upon the Republic of Korea by forces from North Korea constitutes a breach of the peace,
Recommends that the Members of the United Nations furnish such assistance to the Republic of Korea as may be necessary to repel the armed attack and to restore international peace and security in the area.
- UN Security Council Resolution 83 (1950) of 27 June 1950
Thus, the military forces of the free world rallied to South Korea under the banner of the United Nations. While many were also American allies, they joined the fighting not out of treaty obligations to the United States, but instead as member states of the United Nations.
In contrast, the Vietnam War was a essentially a prolonged counter-insurgency operation within the territories of South Vietnam. There was no treaty to oblige American allies to participate in such a conflict. Nor was there any call to arms by the United Nations for defending South Vietnam.
The United States could ask and apply pressure on her allies for help.
Nonetheless, with the lack of any real legal obligation, it fell to each government to decide whether participation was worthwhile. New Zealand, for example, was reluctantly pressured into sending a token force. Great Britain resolutely refused to officially participate, but sent assistance covertly - British soldiers ended up fighting in Australian and New Zealand units, for instance.
French arms had already fought in Vietnam before, and were bloodied by it. I don't think French public opinion could have supported a second trip to the quagmire especially now that they had no empire to fight for in Indochina.
Moreover, French-American relations were not the best during the 1960s. Around the same time the Untied States began committing significant combat troops, France was in fact withdrawing from NATO.