Official issues

The good news is that France and the USA have a bilateral agreement. This agreement ensures that your work history in both countries is considered when applying for retirement benefits. It also clarifies who qualifies and which country pays your pension in different situations. The question, then, is whether you meet the conditions of the agreement.

Looking at your situation, since you’ve worked in France for 9 years and the rest of your career in the USA (assuming minimal career gaps), you’d likely qualify for pensions from both the French and American social security systems when you reach the minimum requirements (credits in the USA and retirement age in France).

The France x USA agreement allows you to combine your work years in both countries to meet eligibility criteria. Without it, you might not have enough credits in either country to qualify for a pension. With the agreement, you can totalize your work years and apply for pensions in both systems.
If the pensions are granted, the French social security will pay for your pension relative to your 9 years working in France (in euros, on a French bank account) and the US will pay for your pension proportionally to your work years in the USA (in USD, on a US-based account).

You must be eligible to retirement in both country to receive both pensions
You must have earned at least 6 credits in the US to apply to the american social security system. For the French social security system, you must be eligible to retirement in order to apply for a pension. That means you must have a total of 166 to 172 trimestres or reach the legal retirment age to request a pension, regardless of how long you worked in either country.

What happens if you have worked in a third country in your carreer?
Things get more complex here (as if they weren’t already!). All agreements relative to social security between France and other countries are bilateral. It means that you may have to pick which biltateral agreement you which to base your pension request on. For instance, say that you have worked in the US for 10 years, in Japan for 5 years and have now been working in France for 10 years. In this case, your best option would be to use the bilateral agreement between France and the USA because it will let you sum up your 20 years of contribution in both countries. In order to make your rights for those 5 years in Japan count, you would have to invoke another agreement (France x Japan or Japan x USA) that will be less attractive.
There are exceptions though. Some bilateral agreements allow for a third country. As I am writing this anwer, such a deal exists between France & Brazil. It lets you include your work time in other countries like Germany, the US, Spain and a few other.