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.