No. The Persian name is a derivation or descendant of the legendary Kay Khosrow. Looking at the list of name bearers Khosrow reveals that the name is in much longer use than 532 CE.
Variants of the name کیخسرو
Husrav, Xusro, Khusro, Khosrau, Khusrau, Chusrau, Khosro, Khosru, Khosrow or Khusraw. In Greek it is sometimes rendered C(h)osroes or Osroes. Arabic it's Kisra (Kisrā) and Turkish Hüsrev. Note that in Kaisar is the Turkish word for Caesar.
The meaning of Khosrow would be "of good standing" or as Wikipedia puts it:
The name Kay Khosrow derives from Avestan Kauui Haosrauuah,2 meaning "he who has good fame".
And on Wiktionary:
From Middle Persian 𐭧𐭥𐭮𐭫𐭥𐭣𐭩 (hwslwdy /Husrōy/), [Book Pahlavi needed] (hwslwb' /Husrōy, Husrav/), from Old Median *Husrava, from Proto-Iranian *Hhuĉráwah ("renowned, famous").
Compare the Encyclopaedia Iranica entry:
KAYĀNIĀN vii. Kauui Haosrauuah, Kay Husrōy, Kay Ḵosrow
The name Haosrauuah is a vriddi formation of *husrauuah “he who has good fame” and ought to mean “good fame” by itself. The later forms, in fact, seem to be descended from *hu-srauuah, although shortening of the initial syllable is possible. The Pahlavi form is usually spelled hwslwy, hwslwd, or hwslwb, all of which should probably be read as Husrōy. The Persian form may derive from this, or it may have been remade in analogy with compounds with -sraw. (On the morphological irregularities of the Avestan name, see Humbach and Ichaporia, 1998, p. 137.)
In the Avesta. The Avesta contains more details about Kauui Haosrauuah (Figure 3) than any of the other kauuis, except Vištāspa. His standing epithets are “stallion of the Aryan lands” (arša airiianąm dax́ iiunąm) and xšaθrāi haṇkərəmō “.?. for command,” where haṇkərəmō has not yet been conclusively interpreted (Yašt 5.49, 15.32). It is also transmitted as haṇkərətō “put together” (Yašt 9.21)
Whereas Caesar is very unlikely to be related.