* Yes, "writing" can be a verb: It's the present participle of the verb "to write," often used in progressive tenses (e.g., "I am writing a story").
* No, "writing" can also be a noun: It refers to the act of writing or the written work itself (e.g., "Her writing is excellent.").
So, to determine if "writing" is a verb or a noun, you need to look at the context of the sentence.