The phrase "appropriate manner" is correct and usable in written English. You can use it to refer to a way of behaving or doing something that is considerate and respectful. For example, "We should always treat others in an appropriate manner.".
Somewhere to celebrate Jeremy's second coming in the appropriate manner.
Terrorists were always wiped out in the final scene, usually in a gruesomely appropriate manner.
The issue of why Mr Saakashvili filed his resignation will be reviewed in an appropriate manner".
They still danced in the appropriate manner when Gangnam Style was played over the public address.
"Opportunity America has behaved in a totally appropriate manner throughout this process," Mr. Schwartz said.
"Should any irregularities be proven, the organisation will deal with those involved in an appropriate manner.
*The physician must end the patient's life in a medically appropriate manner.
Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.
Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com