Which is the proper adjective?
First, we had Nancy Pelosi taking over as Speaker of the House, then Sarah Palin as the Republican vice presidential nominee, and now Hillary Clinton as the presumptive Democratic presidential nominee. So the question of word choice is rising again—Is Hillary Clinton the first female presidential nominee from a major party or the first woman nominee?
Before I answer the question, I want to address a related issue, which is that sometimes it's sexist to point out people's sex because doing so implies that they aren't in their proper role. For example, saying someone is a male nurse or a female doctor wrongly implies that it's so unusual for men to be nurses or so unusual for women to be doctors that you have to make a big deal out of it. But, given that Hillary Clinton actually is the first woman to ever be a presumptive candidate for president of the United States from one of the two main parties, it’s not sexist to talk about it because it is, in fact, a big deal. It’s historic. It’s an important part of the story.
'Woman' as an Adjective
So then, what is the best way to talk about Hillary Clinton being a woman? The words woman and man are primarily nouns, and to say someone is a woman nominee is placing woman in an adjective position. I checked four different dictionaries, and two don't include woman as an adjective (1, 2), one does (3), and the fourth said that when woman is used in the adjective position it's actually an appositive noun and it’s in the process of becoming an adjective (4). So the dictionaries don't give us a clear, definitive answer.
The 'Man' Test
Testing the validity of the sentence by seeing how it sounds to substitute the word man for woman seems like a good way to see if the sentence makes sense. To me it sounds terribly awkward to say someone is the first man nominee. I imagine most of you would say He's the first male nominee, if the need arose.
So, even though some sources say it's grammatically correct to use woman as an adjective, my opinion is that you should say Hillary Clinton is the first female Democratic presidential nominee.
If for some reason the word female makes you uncomfortable, you can use woman as a noun, saying she is the first woman to be the party’s presidential nominee.
With a perfectly acceptable adjective like female available, I don't see any reason to push the word woman into the role. But what about female as a noun?