The bald eagle was not chosen to be the national bird of the United States. It as chosen to be the supporter of the coat of arms of the United States of America when the great seal was adopted.
The reason why the bald eagle was chosen to be the supporter in the achievement of arms of the United States of America when the great seal of the United States was designed remains speculative.
At that time the obverse (front side) of the great seal design of a ruler or government usually had the full achievement of arms encircled by an inscription with the title(s) of the ruler or government.
An achievement of arms would include the "coat of arms" itself on a shield or cartouche. The coat of arms would have a colored field, often partitioned into 2 or more different colored areas, and usually one or more heraldic animals or objects in one or more colors contrasting with the color(s) of the field.
A minimal achievement would have a coronet of rank or crown or a helmet on top of the shield or cartouche with the coat of arms. Helmets were usually covered on top and the sides with a cloth mantling. The mantle might extend down around the 2 sides of the shield or cartouche and have an elaborate leaf like scalloped edge. A crest would sit on top of the helmet and mantle, based on physical crests that had been worn in battles and tournaments in the middle ages. In the 18th century crest were often designed that would have been impractical to physically make, and often floated in air above the coat of arms.
It was and is common to have a motto or war cry written on a scroll or ribbon placed above or below the coat of arms.
Most important persons and institutions had supporters, usually 2 persons or animals, one on the right side of the coat of arms and one on the left side. Some coats of arms had one, three, or four supporters. A pair of supporters, the most common number, would usually stand on a piece of ground or scroll work called a compartment.
Some important nobles used what are called heraldic cloaks as a background for their achievements of arms, and a few monarchs used pavilions as backgrounds for their achievements. Pavilions were like tents with domed tops. Except for occasional heraldic cloaks and pavilions, achievements did not have any further background colors or patterns.
Here is the description of the great seal from the resolution in 1782:
Paleways of thirteen pieces Argent and Gules: a Chief, Azure. The Escutcheon on the breast of the American bald Eagle displayed, proper, holding in his dexter talon an Olive branch, and in his sinister a bundle of thirteen arrows, all proper, & in his beak a scroll, inscribed with this Motto. "E pluribus unum".–
For the Crest
Over the head of the Eagle which appears above the Escutcheon, A Glory, Or, breaking through a cloud, proper, & surrounding thirteen stars forming a Constellation, Argent, on an Azure field.–
A Pyramid unfinished. In the Zenith an Eye in a triangle surrounded with a glory proper. Over the Eye these words "Annuit Coeptis". On the base of the pyramid the numerical letters MDCCLXXVI & underneath the following motto. "novus ordo seclorum"
Technically the coat of arms of the USA is white or silver, with six thin red vertical bars, and at the top a blue area from side to side. Since this is on a shield on the breast of the eagle, and the background behind the eagle does not have a specified color or pattern, the eagle is the single supporter of the coat of arms. The eagle is part of the achievement of arms that includes the coat of arms, not part of the coat of arms.
Note that the cloud and glory are usually depicted as a circle, like the halos or glories around the heads of the imperial eagle of the Holy Roman Empire. The cloud and glory are described as a crest over the head of the eagle. In those days many crests were depicted floating freely above the rest of the achievement of arms. The motto in a scroll held in the eagle's beak is connected to the rest of the achievement better than most mottos are.
The achievement of the USA does not have a compartment, or helmet, or mantle, or any crown or coronet, nor any heraldic cloak or pavilion.
So what possible inspiration could there be for the eagle and its design features? Choosing a bald eagle is easy to explain once an eagle has been selected, since the bald eagle is a type of eagle native to the Americas.
In 1782 the members of congress would have some familiarity with the arms, seals and heraldic achievements of the powers of Europe. Those powers included the Papal States - a catholic theocracy unpopular with most of the founding fathers - three republics, and less than a dozen monarchies.
The three republics included Venice, a centralized state ruling many colonies like Great Britain had, and the decentralized Swiss Confederation and United Netherlands that would have been more inspiring to the founding fathers especially during the time when the loose Articles of Confederation seemed to be strong enough. I do not see any heraldic inspiration from them except for the bunch of arrows in the sinister claw of the American eagle being based on the bunch of arrows in the sinister paw of the lion of the States General of the United Netherlands.
There are examples of eagles with objects in both claws. At that time the Russian imperial eagle held a scepter in the dexter claw and an orb in the sinister claw, the eagle of the Holy Roman Empire held both a sword and a scepter in the dexter claw and an orb in the sinister claw, and the eagle of the kingdom of Prussia held a scepter in the dexter claw and an orb in the sinister claw.
Eagles with coats of arms on shields on their breasts were common in the Holy Roman Empire. The loose organization of the Holy Roman Empire, with most powers in the hands of the various states, and the monarch being elected by an electorial college, would seem more inspirational to the founding fathers than most European monarchies.
The (elective) emperors began putting the coats of arms of their hereditary possessions on the breasts of the imperial eagle about 1440. That is an example of the imperial eagle in the gold or yellow field of the imperial coat of arms having other coats of arms on its breast. There were also many examples of the imperial eagle being used as a single supporter of other coats of arms. The coats of arms of many princes and many prince bishops had their territorial arms on the breasts of the imperial eagle.
And prince bishops, prince archbishops, and prince abbots, were elected by small bodies of clerics in their domains, thus somewhat resembling the governors of more conservative American States.
There were many city states in the Holy Roman empire, republics that did have some resemblance to the governments of some of the more conservative American states. Many of them put their coats of arms on the breast of the imperial eagle either as a smaller coat of arms on top of the coat of arms of the empire, or else using the imperial eagle as a single supporter for their coat of arms.
So the founding fathers could have seen an eagle as an emblem of the civic virtues of the ancient Roman republic. And in the present day world they could see that the eagle was used as the coat of arms of the Holy Roman Empire that claimed to be the rightful successor of the Roman Empire and had some features that they could admire.
The United States was sometimes described as a new empire even back then, since it had a very large territory like the Holy Roman Empire, and was composed of states with large power relative to the central government like the Holy Roman Empire. The Holy Roman Empire with empire-level decisions made jointly by the elected emperor and the Diet consisting of representatives of the various states operated similarly to a decentralized republic like the Swiss Confederation, the United Netherlands, or the United States under the Articles of Confederation.
So the founding fathers might have seen the Roman eagle as not only a symbol of the bygone civic virtues of the ancient Roman Republic, but also of a large modern realm where states with large autonomy and a large degree of freedom from the central authority joined together for common purposes in a largely republican manner.
Added May 9, 2018. Wikipedia does claim that the bald eagle is the national bird of the United States.
And it gives this as the source:
So far I have not found any reference to the bald eagle being declared the national bird of the USA on any specific date. Sources often give the adoption of the great seal of the USA in 1782 as the date, but that is the date that the bald eagle became the supporter in the US coat of arms.