Long before the first Valentines Day,
I was the narrow standard.
Herpes has been known for at least 2,000 years. Also, Valentines Day = VD = venereal disease (clued by "first") OP -> abbrev for standard (clued by "narrow") = STD
And while I'm neither alive nor dead,
all who meet me are branded.
"Are viruses alive?" is a debated topic. Herpes carriers are often "branded" by visible sores.
Once met, you can't hide in fire or ice;
plus I'll come on your worst of days.
People try to treat the symptoms with hot and cold compresses. Sores often appear during times of sickness, hence their nicknames "cold sores" and "fever blisters". Sores can also be triggered by stress.
Then defeated I wait for my next chance to strike,
and rise like the phoenix in flames.
The symptoms go away and come back. OP -> this is known as a "flare up" (clued by "rise" and "flames")
The unhappy marriage of complex and simple,
I am a tough itch to scratch.
The virus is known as "Herpes simplex", though there are complexities in its structure (i dont know, the word "complex" shows up a bunch in the wiki page). OP -> SIMple + comPLEX = SIMPLEX.
And once you steal me from her garden,
you'll want to give me right back.
OP -> once a person attains this things, they do not want to have it anymore. "Garden" just helps tie the vegetable clue together
I'm her least favorite vegetable