I'm not an attorney, but I wanted to get to the actual laws regarding the Charlie White case because I found the press coverage confusing. I do that by starting with the the story as reported in the web edition of the Evansville Courier & Press

From the newspaper, paragraph 3.

But the governor said he was holding off on naming a permanent replacement because a judge could reduce the charge to a misdemeanor, allowing White to regain the office.

This is a correct summary of the statement the Governor released. But it ignores Indiana Code 3.8.1.5(d) , which states:

The subsequent reduction of a felony to a Class A misdemeanor under IC 35-50-2-7 or IC 35-38-1-1.5 after the public announcement of a felony... does not affect the operation of subsection (c).

This section of the law describes candidacy and not a condition for an officeholder who is convicted of a felony. There is a separate statute (IC 3.5.1.38) for felonies committed while in office which appears to allow for Mr. White's return.

However, because at least one of Mr. White's felonies was committed before taking office, one can certainly argue that he could have been a convicted felon prior to taking office if the criminal justice system had worked more quickly. This would become a question for the courts, but only if the Hamilton County Superior Court were to downgrade the felony convictions to misdemeanors.

The second legal issue presented in the article relates to the assertion of Democratic state chair Dan Parker that Democrat Vop Osili should take office. I understand that this is what was ruled by Marion County judge Louis Rosenberg on multiple occasions, most recently on December 21, 2011. This is accounted for in IC 3.12.11.25(a)

Except as provided in subsection (b), whenever the commission makes a final determination under section 18 of this chapter that the candidate who is subject to a contest proceeding is not eligible to serve in the office to which the candidate is nominated or elected, the candidate who received the second highest number of votes for the office is entitled to a certificate of nomination or certificate of election even though a certificate may have been issued to another candidate upon the tabulation of the votes.

However, I reference IC 3.8.8.8(b) and (c) which seems to contradict 3.12.11.25:

(b) If, after the election, it is determined as provided by law that the individual was not qualified to be elected to the office, it shall be considered that: (1) an eligible candidate of the same political party, if any, as the ineligible candidate had been elected; and (2) a vacancy in the office occurred after the election. (c) The vacancy in the office shall be filled as otherwise provided by law.

If this is the applicable section, then the Governor would appoint Mr. White's permanent replacement. The ambiguity is somewhat removed by the section title of 3.8.8.8: "Effect of candidate who withdraws receiving most votes in election; filling vacancy" (emphasis added).

Because there is at least the potential for ambiguity, I believe this case will end up with a decision by the Indiana Supreme Court, and I suspect it would lean towards the Governor appointing White's replacement.