That's easy-- no, under current canon law:
Abstinence from meat, or from some other food as determined by the Episcopal Conference, is to be observed on all Fridays, unless a solemnity should fall on a Friday. Abstinence and fasting are to be observed on Ash Wednesday and Good Friday (Can. 1251).
In the new calendar, the Feast of the Annunciation is a Solemnity, as was the Feast of St. Joseph, for that matter.
This is confirmed by the Lenten regulations published by Archbishop Carlson in the Review:
All other Fridays of Lent, with the exception of the Feast of the Annunciation, March 25, are days of abstinence from meat.
Simple enough, eh? But this question does have relevance for those Catholics who choose to adhere to the traditional fasting and abstinence disciplines of the Church, and who are attached to the Extraordinary Form of the Roman Rite, which retains the traditional calendar. In the traditional calendar, the Feast of Annunciation is a First Class Feast, as was the Feast of St. Joseph, for that matter. Does this equate by analogy to a "solemnity", as both terms indicate the highest level feast possible? It sure seems so. Hence, problem solved, eh?
Well, maybe not. Under the Code of Canon Law of 1917, which was the one in effect prior to the change of the calendar and the subsequent allowance to lift the abstinence requirement on a "solemnity", the key to whether a feast obviated the abstinence (and fasting, also then in force) duty was whether it was a holy day of obligation. In other words, it wasn't enough that it was a first class feast (or solemnity, if you will), it had to have been a day of precept:
Canon 1252 (CIC 1917). 1. The law of abstinence only must be observed every Friday.
2. The law of abstinence together with fast must be observed every Ash Wed, every Friday and Saturday of Lent, each of the Ember Days, and the vigils of the Pentecost, the Assumption of the God-bearer into heaven, All Saints, and the Nativity of the Lord.
3. The law of fast only is to be observed on all the other days of Lent.
4. On Sundays or feasts of precept, the law of abstinence or of abstinence and fast or a fast only ceases, except during Lent, nor is the vigil anticipated; likewise it ceases on Holy Saturday afternoon.
As you can see, the law used to bind Catholics to abstain every Friday of the year (and still does, or some other penance, but that is a different post), and also on Saturdays of Lent as well. It also required fasting on every other day of Lent, except Sundays. So, it would seem that if one wished to keep the formerly-binding rules of fasting, it would still have been required on the Annunciation because a) it is not a day of precept in the U.S., and b)it occurs during Lent.
So, problem solved, eh?
Well, maybe not. The liturgical calendar in existence at the time of the promulgation of the 1917 Code of Canon Law was more complex than the more simplified one later in use from time of Blessed John XXIII, with varying subcategories of feasts. And some of the Vigils were suppressed, that would have affected fasting. But that doesn't affect our question of the day, as the Code lifted the abstinence and fast requirement only on days of precept outside of Lent.
And, if you want to get all technical, and be super-duper abstemious, it used to be the rule long ago that you had to fast and abstain every day of Lent, and even to have modified fast during Septuagesima onwards. And the abstention from meat encompassed not just meat but eggs, dairy and basically anything that came within 50 feet of a cow.
I go into this detail only because it comes up each Lent among those who try, from motives of personal piety, to undertake a more traditional Lenten fasting discipline.
The bottom line is that the current canon law is clear, and that is all that binds in conscience. So you can eat meat this Friday, and you didn't have to fast anyway. And, thinking like a lawyer, since the Extraordinary Form is entitled to use the traditional calendar, the question of the Annunciation is a valid one. But, using current law applied to traditional feast categories, it sure seems like the same result is reached. If you really want to do it the "traditional way", you can abstain and fast, but that is your decision, not mandated by law.
If you think that was a little complex, don't get me started on the Communion fast.
And if you are still reading this, I'll see you Sunday at St. Francis de Sales.