To Make a Long Story Short 3 June 1784 is an adequate date for the formation of the United States Army - meaning the standing or regular army of the USA. The USA had and has other armies which are not The United States Army.
The Long Story
In colonial times the colonies had a militia military system. Every able bodied man within the legal age limits was legally required to own a gun and ammunition and to attend the periodical training sessions of his local militia company. In times of Indian or other conflicts the colonial government would call out the militia and mobilize it for combat.
In some of the later colonial conflicts the colonies would also raise units of volunteers to be full time soldiers for the duration of the conflict, and sometimes the British government sent regular army units of professional soldiers to fight in the colonies.
During the American Revolution the majority of the fighting men were members of the militia units. But because of problems with using only militia the Continental Congress established the Continental Line, which became a standing or regular army, beginning on June 14, 1775.
When the Continental Army was demobilized in 1783, a single regiment, the 1st American Regiment remained, under the command of Colonel Henry Jackson. In 1784 this regiment was disbanded. A single company of artillery was reassigned to a new regiment, the First American Regiment, the predecessor to the United States Army's 3rd US Infantry Regiment.
So each state in the United States had its own military force, its militia of part time soldiers, which could be mobilized in time of war.
In the generations after the American revolution, the militia system began to decay, and the militia became less and less effective, until it was finally reorganized as the National Guard under joint federal and state command in 1903 and 1933.
The United States National Guard, also commonly referred to as just the National Guard, is part of the reserve components of the United States Armed Forces. It is a reserve military force, composed of National Guard military members or units of each state and the territories of Guam, the Virgin Islands, Puerto Rico, and the District of Columbia, for a total of 54 separate organizations. All members of the National Guard of the United States are also members of the militia of the United States as defined by 10 U.S.C. § 246. National Guard units are under the dual control of the state and the federal government.
National guard membership is part time until units are mobilized.
The regular army was formed on June 3, 1784, with the formation of the First American Regiment, which became the First Infantry in 1791. In 1815 it was merged with other regiments to form the Third United States Infantry, the "Old Guard" of the US army.
The Legion of the United States was formed in 1792 out of existing regular army units and new recruits. It contained four sub legions each having 2 infantry battalions, 1 light infantry battalion, 1 cavalry company, and one field artillery battery.
The Legion of the United States was abolished in 1796, the 1st, 2nd, 3rd, and 4th sublegions becoming the the 1st, 2nd, 3rd, and 4th infantry regiments of the US Army.
During the Northwestern Indian War the US also raised units of levies, temporary units of volunteer soldiers. Richard Butler, major general of levies, killed in St. Claire's Defeat, or the Battle of the Wabash, in 1791, was the highest ranking US officer ever killed by Indians.
During 19th century conflicts, mostly the War of 1812 (1812-15), the Mexican-
American war (1846-48), the Civil War (1861-65), and the Spanish-American war (1898) the major United States fighting forces were units of volunteers. Men would volunteer to become full time soldiers in temporary units raised to fight in the war and be disbanded afterwards.
Must volunteer units were raised by the states and territories, and thus had designations like 49th Pennsylvania Infantry or 1st Indiana Heavy Artillery or 7th Michigan Cavalry, etc., etc. Sometimes volunteer units were paid and equipped and under the command of the states and territories that raised them, but most of the volunteer units would be organized by the states and territories and then mustered into federal service to be paid, equipped, & supplied by the federal government and under the command of the federal government.
The federal government also raised volunteer units without any state or territorial affiliation, such as units of sharpshooters, and units of Rebel prisoners, and the United States Colored Troops during the Civil War.
Volunteer units often elected their officers, who were commissioned by their state or territorial governors, up to the rank of colonel. All generals of volunteers were commissioned by the federal government, and are sometimes distinguished from regular army generals by the initials USV (for United States Volunteers) instead of USA (for United states Army).
Among officers of the same rank, regular army officers were senior to volunteer officers who were senior to militia officers.
During the 19th century wars, both the United States Volunteers and the United States Army were paid, equipped, supplied, fed, and commanded by the federal government and were federal armies, despite the state and territorial designations which most volunteer units retained to satisfy state pride.
Even though militia training was legally compulsory, after the American revolution the militia laws became a less and less enforced and more and more men ignored them, and in many locations the militia became more or less social clubs for men who wanted to occasionally drill and wear fancy uniforms.
The regular army was always recruited by voluntary enlistment for a term of years, and the United States Volunteers were obviously volunteers. Conscription was first used during the Civil War, and not again until World War One. During the Civil War only a small minority of federal recruits were conscripted or paid substitutes for conscripted men.
Therefore, the use of "drafted" in the original question implies a belief that the 19th century federal and state governments were far more tyrannical than they were.
Some people would say the United States Army dates back to 1796, but that was just a change of name after the Legion of the United States (1792-96) was disbanded.
(added 04-11-2019) The federal government under the present constitution began functioning and the constitution came into effect March 4, 1789. So as sempaiscuba suggested, some persons may consider that the federal government began to exist and have a federal army on March 4, 1789.
The official foundation date of the federal government's standing army, the United States Army, is June 3, 1784. The Third Infantry, the "Old Guard", dates to then.
(added 04-11-2019) The Articles of Confederation creating a weak central government for the USA came into effect on March 1, 1781, when ratification was completed. Thus there was an officially created central government to have a central army since March 1, 1781.
(added 04-11-2019) the Continental Congress approved the Articles of Confederation creating a weak central government on November 15, 1777 and began submitting them to the states for ratification. The Continental Congress used his powers under the Articles of Confederation during the years it took for them to be ratified, so an ad hoc central government existed since November 15, 1777.
The First Battalion, Fifth Field Artillery, is the oldest unit in the regular army, tracing its lineage back to a unit of the Continental Line founded January 6, 1776.
(added 04-11-2019) The Continental Congress appointed a committee to plan a union of the states on June 12, 1776, thus beginning the process that eventually resulted in a central government of the USA.
The first professional army in the United States and under the control of the Continental Congress, the Continental Line, dates back a little farther to June 14, 1775. But the Continental Line was all disbanded, except for what became the First Battalion, Fifth Field Artillery, in 1783 and 1784.
Many National Guard units date back to colonial times, the oldest being four units of the Massachusetts National Guard tracing their lineage back to 1636.