After a bit more research, I'd rate this true but misleading.
Here is the Consumer Price Index for July 2022
As highlighted in the screenshot, the month-over-month inflation for July 2022 was indeed 0%. This means that the weighted-average of prices measured by the CPI was exactly the same in June 2022 and July 2022.
However, "inflation" is usually given as a 12-month change of price-averages, not a 1-month change. Since the 1-month inflation was positive for 11 out of the past 12 months, it should be no surprise that the 12-month inflation is also positive, at 8.5%.
In other words, while month-over-month inflation was indeed 0% for July, what would normally be called "the inflation as of July" was not.
The claim by Fox news that 0% refers to the change in inflation is straight up false. Both the month-over-month and year-over-year inflation values went down, meaning the change in inflation is negative in both cases.
Presumably Fox is confused because the above chart says "adjusted changes from preceding month". However this refers to the change in the CPI, which is what defines inflation, not to the change in inflation itself.
As an aside, they've now made this claim in two articles, neither of which mentions the month-over-month vs. year-over-year distinction, despite the second being a critique of a NYT article which explains the distinction. That makes it hard to pass this off as an innocent mistake.