First thing to note it was highly unusual for explorers to name the new lands after themselves. In fact, up to that point, it was also unusual to in general name lands after the explorers who found them, although exceptions existed, and the practice would soon change. Mostly because it became simpler to make it clear what geographic region you are referring to by mentioning the explorer who found it, rather than generic geographical names and features, or Saint's names which predominated the naming conventions up to that time. It's also worth mentioning that names of places quite often changed in the early times after discovery, before a convention would arise, often contradicting the original name (e.g. Brazil was named by Cabral as Vera Cruz, to be immediately renamed by King Joao II as Santa Cruz, for it then to become solely referred to as Brazil soon after).

Second thing to note, is that Vespucci didn't name the continent of (South) America after him. He didn't name it at all actually. In his letters to Medici (either the widely printed version supposedly from 1503 titled sometimes just Mundus Novus which some think was fake/forgery, or the alternatively proposed original letter from Cape Verde from 1501/1502 ) only ever uses the phrase "new world" or "mundus novus" in referring to the landmass he 'found'.

The person who first proposed the name America (as it is now commonly held at least) was German Cartogropher Martin Waldseemüller in his work on geography Cosmographiae Introductio from 1507 which was also published with his maps, and also Latin version of Vespucci's Four Voyages (it's worth to note that today some historians also think the Four Voyages work was a forgery and not an actual Vespucci's work, just published under his name by someone taking fragments from his actual letters and adding / making up parts frivously)

To get back to Cosmographiae Introductio, in his work Waldseemüller proposes the name America and is specific in offering why. His original reasoning in Latin can be read here, page XXX while the english translation can be found in the same link, page 70. I'll copy paste the translation:

Now, these parts of the earth have been more extensively explored and a fourth part has been discovered by Amerigo Vespucci (as will be set forth in what follows). Inasmuch as both Europe and Asia received their names from women, I see no reason why any one should justly object to calling this part Amerige, i.e., the land of Amerigo, or America, after Amerigo, its discoverer, a man of great ability. Its position and the customs of its inhabitants may be clearly understood from the four voyages of Amerigo, which are subjoined.

So, Waldseemüller reasoning is such: Vespucci discovered this 'fourth' part of the world and it should be named after him. As the old world continents like Europa and Asia were named after women (in previous paragraphs Waldseemüller claims Asia was a queen there, and Europa a girl from Greco-Roman mythology) he proposes a feminine version of the name Amerigo or Latin Americus, - America - to match the other feminine names of Europa, Asia, and Africa (or Libya, as Africa was alternatively named by ancient writers, as Waldseemüller notes).

Now, Waldseemüller also leaves us some gaps in understanding his reasoning. Why he chose Vespucci's name over surname is one such. I've seen theories that he wanted to keep the name beginning with vowel (America, Europa, Asia, Africa) but as far as I can tell there is no evidence to back it up. He might just as well thought it was prettier. Adding more confusion, other strange things with his reasoning appear. Namely, why he used Latinised form Americus for Amerigo's name, as the true Latin form of his name is Albericus, and indeed, prints of Vespucci's name in Latin often appear under correctly latinized name of Albericus Vespucius or variants (examples: 1504 edition of Latin translation of Mundus Novus, with the writer being attributed to "Albericus Vespucius". And 1506 German translation, also attributed to "Albericus Vespuccius"). Unfortunately, we can only guess Waldseemüller thoughts at the time about those issues.

However to recapitulate: Vespucci didn't name anything after himself, others did. Specifically Martin Waldseemüller in 1507. He says he goes with America to pay homage to the discoverer (at least who he believed was the discoverer) and chose the feminine form to be more matching to the existing continent names.

Amerigo Vespucci did not name those lands. The name America first appeared inMartin Waldseemüller's map of 1507, which was the first printed map that contained the lands of the New World. This begs the question on why did Waldseemüller name those lands as such.

The cartographic information on the Indies was relatively scarce up to that moment. The only detailed maps were possessed by the Casa de la Contratación de las Indias in Seville, and in the Royal Court of Portugal in Lisbon. The Casa de la Contratación kept the most accurate maps possible, and recceived information every year from the expeditions sponsored by the Crown and the Casa, updating the Padrón Real (Royal Census, id est the general map of the New World) with every new information. This updating was the duty of the Piloto Mayor, the chief cartographic officer of the Casa de la Contratación. In 1506, the Piloto Mayor was none other than Amerigo Vespucci, who obtained the post under recommendation of Christopher Columbus' himself, who always cared about Amerigo for they both were foreigners, Italians specifically. Amongst the duties of the Piloto Mayor was to keep the information as state secret, as the information on the new lands was of vital importance to the Crown.

Not quite exactly fulfilling his duties, Amerigo Vespucci sent some cartographic information to the German cartographer Martin Waldseemüller, who had requested some info from the Casa in order to create a world map. The maps sent by Amerigo to Martin were not the most recent or most updated maps he had, of course. If you carefully analyse Waldseemüller's map you can spot some interesting details like the shape of Cuba, that closely matches the one depicted in Caverio's map, which is a copy of the anonymous map of Alberto Cantino from 1502. The coastline close to Cuba that goes on a South-North axis is not Florida as you could deduce from the shape, but it's the northern coast of the Yucatán peninsula. This also comes from a map related to the Cantino and Caverio maps, as evident from analysis.

Even if the information was not the best, or the most up to date, Waldseemüller thanked Amerigo Vespucci by placing his name on the New World, and even depicting him alongside Ptolemy on the upper part of the map. Waldseemüller would change his mind in 1513, when he made his Universalis Cosmographia, not having the name of America on it anymore, but it was late, the name had already caught on despite the Indies denomination being more frequent.

As for the -ia suffix, it is the common suffix for place names in Latin, which was the universal language of culture until the XVIII century. Think of some Latin place names and you'll come to that realisation: Hispania, Gallia, Lusitania, Italia, Britannia, Graecia, Dacia, Germania, Rhaetia, Moestia, etc.