You are right that the French Open is the only Grand Slam tournament without Hawkeye,
The reason is that Clay, unlike grass and hardcourts, leaves a mark that players and umpires can check and verify if the ball was in or out.
You can argue that some mistakes can be made but that's quite rare, A hawkeye system is very expensive and the cost/benefit ratio doesn't pays.
Gilbert Ysern, the director general of the French Open and a former umpire, address to this:
“I don't think we need it,” he said in an interview in his office.
“There are ball marks on clay,” Ysern said with a genial smile, “and our chair umpires are used to checking the marks when needed, and, so why would we need Hawk-Eye?”
And the disputes?
“It happens very, very rarely that the officials can't find the mark,” he said.
Source
sports.stackexchange.com/users/385/Dor Cohen
I strongly suspect that the manufacturers of Hawkeye and similar ball tracking systems would actually prefer not to have a hard 'ground truth' to compare to being beamed live to millions of viewers.
These systems are not as accurate as the TV graphics imply and there use as part of the umpiring process is not all to do with accuracy.
It is as much about having a non-human, dispassionate system
have the final say when the humans can't agree as arguing with a machine is futile.
The manufacturers of Hawkeye claim a mean accuracy of 3.6mm, although this figure, or at least its interpretation has been called into question .
Basically, since this is an average over some range of shots, it is likely that the actual error in certain circumstances is significantly higher (as detailed in the second reference).
Errors approaching 10mm could easily turn the correct result, as observed by the skid mark on clay ground truth, into an incorrect Hawkeye call.
It would be terrible publicity to have a top player angrily pointing to the clear evidence on the court that differs from the Hawkeye prediction.
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Clay courts leave a visible mark which shows where the ball lands.
This is the reason why the staff at Roland Garros clear the court often during a match so there wont be overlap of marks(or mistake of the actual mark), which can cause confusion.
If a ball is questionable, then the referee will come down and check the mark on the court and make his decision about whether it was in or out .
sports.stackexchange.com/users/1469/Simon Yundov
Balls can skid much longer distances on clay.
On hard courts only the ball measurably changes (deforms) on impact.
On clay both the ball and the court are changing simultaneously as a layer of dirt is shifted around.
These factors make Hawk-Eye's job more challenging on clay and it would be logical to conclude that Hawk-Eye's margin of error increases on clay.
sports.stackexchange.com/users/1436/dimalinux
Hawk Eye is in general more accurate than the human eye
but that doesn't mean it's 100% perfect.
It will always remain an approximation of the "ground truth".
With clay courts we have the unique ability to see the actual physical mark.
This hard, physical evidence should be always preferred to approximation, no matter how accurate.
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I have never seen Hawk-Eye technology to be used on clay tournaments. French Open is running at the moment and it is not being used there.  What are the reasons for not using this technology on clay? Do tennis federations have some rules against it?
Why is Hawk-Eye not used on clay courts
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The reason is that Clay, unlike grass and hardcourts, leaves a mark that players and umpires can check and verify if the ball was in or out. There is always a margin of error with technology and it would be embarrassing for the manufacturers if the technology called differently from the actual evidence in the clay. Unlike grass, clay leaves a visible mark. Mistakes are rare, therefore the cost of a hawkeye system is not justified. You might also want to consider that the Hawkeye system is only an approximation of hard evidence. Since clay leaves a visible trace, Hawkeye is not as effective, and therefore redundant. More to the point, some people argue that ball tracking systems like Hawkeye are not actually objective, so possible clash with visible evidence, such as clay, may cause bad publicity. Finally, the Hawkeye system is not as effective on clay, given that balls go further and may deform on impact, while court conditions might change.
Unlike grass, clay leaves a visible mark. Mistakes are rare, therefore the cost of a hawkeye system is not justified. Some people argue that ball tracking systems like Hawkeye are not actually objective, so possible clash with visible evidence, such as clay, may cause bad publicity. The Hawkeye system is not as effective on clay, given that balls go further and may deform on impact, while court conditions might change. You might want to consider that the Hawkeye system is only an approximation of hard evidence. Since clay leaves a visible trace, Hawkeye is not as effective, and therefore redundant.