The -y ending is a standard transformation from Latin to English, where Latin names in -ia, -ius become English -y: Livius > Livy, Aemilia > Emily, Italia > Italy, Lucia > Lucy, Antonius > Antony, etc. This is standard for names that came to English from Latin.

In this context, the written form July is an absolutely standard transformation.

But the pronunciation, with y pronounced as in sky rather than as in Italy, is anomalous. The shift in vowel is caused by a shift in syllable stress.

In English, words with final -y vary between the sounds /i/ and /ʌɪ/ depending on whether the final syllable is stressed or not. Other names derived from Latin -ia, -ius have their stress before the last syllable: Italy, not Italy. This also goes for personal names derived from Iulius -- Julie, July, Julius, Julia -- which have their stress on the first syllable, and which have the /i/ sound in the relevant position.

By contrast, words where final y has the other sound, /ʌɪ/ as in sky, have it in a stressed syllable: deny, apply, awry, words ending in -ify (amplify, signify, etc.).

The month-name July also had its stress on the first syllable up to the 1700s, according to the OED:

The word was usually stressed on the first syllable in the early modern period, as the form July-flower, due to folk etymology (see γ forms at GILLYFLOWER n.), implies. The orthoepists Peter Levins (1570) and Elisha Coles (late 17th cent.) both include the word among those which have unstressed -y, and Johnson (1755), W. Johnston Pronouncing & Spelling Dict. (1764), and J. Walker Dict. Answering Purposes of Rhyming (1775) all indicate stress on the first syllable (Johnston also marking the y as 'long'). Both occurrences of the word in Shakespeare are so stressed, as are most metrical examples down to the late 18th cent. (compare quots. 1704 at α., 1736 at α., 1781 at Compounds 1). Stress on the first syllable still occasionally occurs in Scotland.

So it looks like the pronunciation inititally shifted from sounding like the personal name Julie, in the 1500s-1600s, to having a more protracted vowel in the second syllable, in the 1700s. The second syllable then acquired the stress. And from there, it was just a matter of applying the standard English pronunciation rule for final -y.

Edit: by the way, July and August weren't added to the calendar, they were renamed. Previously they were called Quinctilis and Sextilis.