As opposed the the Argentine myth of an Argentine state existing all the time since 1810 (or 1816), here is a very brief overview.
In 1820 began more than 40 years of civil war, where the united provinces were not united.
- - -
In what is today the state of Argentina, as late as September 1859 there were:
two states (La Confederación and Estado de Buenos Aires) - there had been others during the years
two capitals (Paraná and Buenos Ayres)
two constitutions (1853 (La Confederación) and 1854 (Estado de Buenos Aires)
two presidents (Urquiza in Paraná and Alsina in Estado de Buenos Ayres)
two senats
two congresses
The war between the Confederation and Buenos Aires lasted nearly a decade until, in the Battle of Cepeda (1859), the Argentine Confederation army defeated Estado de Buenos Ayres's army, following which Mitre ultimately abrogated the Pact of San José, leading to renewed civil war.
These hostilities culminated in the Battle of Pavón (1861), and to victory on the part of Bartolomé Mitre and Buenos Ayres over Urquiza's national forces. President Santiago Derqui, who had been backed by Urquiza, resigned and the Argentine Confederation was replaced by the Argentine Republic on 17 December 1861.
Spain did not accept Argentina’s independence until 9 July 1859, when the first “Tratado de Reconocimiento, Paz y Amistad Firmado por la Confederación Argentina con S.M. la Reina de España” was signed in Madrid.