If I could choose neither, I would, since I'm not sure if the sentence is grammatical (I have asked a separate question about that here: Is "I am who(m) God made me" grammatical?).
If I had to choose one and I was allowed to choose based on my own preferences, I would choose who since, as many posts on this site explain, "whom" often sounds stuffy or pedantic.
But if I had to choose one based on what I think fits best with prescriptive grammar, I would choose whom.
This isn't a complete answer, so much as a summary of what I've discovered while researching this question. I hope some syntax expert will post a more a definitive answer.
Syntax: the "object complement"
In sentences of the form "[God] [made] [me] [a man]," "God" is the subject, "made" is the verb, and "me" is the object of the verb. The phrase "a man," rather than being an object, serves the role of what is called an "object complement."
The grammatical case of the object complement
This is tricky to ascertain, since most sentences with a pronoun in this position sound very awkward and unnatural ("God made me her"?/"God made me she"?).
However, I think there are several fairly strong arguments for it being in the objective case.
For one thing, I was able to find one fairly common expression that uses this structure with an objective-case pronoun as the object complement: "what makes me me" and variations on this phrase. As far as I can tell, no one say "what makes me I," and my intuition tells me that this would be ungrammatical.
Unfortunately, this is not a foolproof argument because online examples that are similar to this often show some odd punctuation that casts doubt on the role of the pronoun here. Often there is a comma or ellipsis before the pronoun: "What made me, me", "everything that made me ... me". In contrast, a comma would be ungrammatical in "God made me, a man." This might be a sign that these phrases have different grammatical structures.
Sometimes, what appears to be the object pronoun is capitalized or put in quotation marks: "What events in my life had made me, Me?", "the things that made me 'me' ". This odd treatment almost suggests the word "me" in this expression is being treated as a noun rather than a pronoun, and in that case it would not inflect for case and would be useless as evidence for the "who"/"whom" rule. (A parallel case: we say "The Me I Want to Be", not "The I I want to Be," but prescriptive grammar still prescribes the nominative form in the phrase "who I want to be.")
But, I think there are also theoretical arguments that suggest the object complement is in the objective case (or at least, that it "should" be in prescriptivist grammar). The most straightforward is that it's a complement, and prescriptivists often value arguments like "complements should match the case of their referent." That's the whole foundation of the argument for saying "It is I" rather than "It is me."
From a more descriptive viewpoint, I have read that the "objective" case or pronouns is generally less marked in modern English, which I think means that phrases tend to default to that unless explicitly assigned to the nominative case by a rule. However, the objective case of "whom" is an exception since it basically only occurs in educated constructions that have to be explicitly taught. Some linguists argue that the rules for using "whom" have actually become distinct in the modern language from the rest of the English case system. (Some papers I found about this that I still have to finish going through: The Who/Whom Puzzle: On The Preservation Of An Archaic Feature, Whom and the English Case System).
Is it grammatical to replace the object complement with a pronoun?
I'm not sure if it's grammatical to replace this element of the sentence with a fronted relative pronoun. (I believe "who" or "whom" in this context would be a relative pronoun, although I'm not sure: it's possible it would be an interrogative). I'm currently trying to research this.
There are some examples of sentences with object complements here: http://www.englishgrammar.org/object-complement/
- They elected Martin their president.
- They named the boy Christopher.
I don't get great results when I try to modify them to use the pronoun "what":
- *What they elected Martin was their president.
- ?What they named Christopher was the boy.
- ?What they named the boy was Christopher.
These all seem awkward to me, although Mitch and Araucaria have left comments indicating that "What they named the boy was Christopher" seems OK to them. The comments that used to exist below this question indicate that many people think "I am who(m) G-d made me" sounds ungrammatical, although I have not found any source that explains why it would be.
Araucaria also found some real-life examples of "I am/ This is who God made me" that I think are worth listing in an answer:
According to typical prescriptive rules, the pronoun's role in the matrix clause shouldn't affect its form
The matrix clause (or main clause) in this example is "I am __." In general, the choice of "who" or "whom" is not affected by the pronoun's role in the matrix clause, only by its role in the embedded clause. So the prescriptivist rule about using the nominative form of a pronoun after "I am" should be irrelevant. (A similar example is discussed in this following question: Who vs. whom in complex sentences—"I gave the prize to whoever deserved it most" is correct, even though we would say "I gave the prize to him/her," because the case is determined by the embedded clause "deserved it most" and we we would say "He/She deserved it most.") The following questions are also relevant: Can a phrase be the object of a clause and how would its subject change?, "I give it to him who came first" vs. "to he who came first", Which is grammatically correct: "Let he who..." or "Let him who...".
For some speakers, there are additional complications in phrases such as "the person whom the police thought was responsible," but as Geoffrey K. Pullum explains in the linked article, the use of whom in this context is generally considered incorrect by prescriptivists.
A caveat: F.E.'s answer to the following question is also relevant, and points out that in real English usage (as opposed to the rules of prescriptivists), "fused relative" constructions like this with clashing case requirements may be avoided:
"Put me in touch with whomever created it"?
Wikipedia cites an interesting passage from "The Distribution of Pronoun Case Forms in English", by Heidi Quinn, that indicates that in a similar construction in Old English (the "argument relative"), the case of the wh-pronoun was based on its role in the matrix clause.
For example, the wh-pronoun wam functions as the subject of the relative clause in (65), but the relative itself functions as the object of the matrix preposition to. Since the matrix clause always wins out over the relative-internal clause, the wh-pronoun in (65) surfaces as the objective form wam, rather than as the nominative form hwa.
Unfortunately, the page with the relevant sentence was not visible in the Google Books, but I found what seems to be the same example in a PDF of Quinn's thesis:
Ðe holi gost ... hine dealeð to [wam him beoð lofue]
the holy ghost ... 3sgM.ACC gives to wh.OBJ 3sgM.OBJ is pleasing
'The holy ghost ... gives it to whomever is pleasing to him.'
(Layamon (Otho) 9081) [AlIen 1980: 208]
I'm not actually sure based on this example if this construction is really analogous to the modern English construction, but it's interesting nevertheless. You can see more discussion on the Wikipedia talk page.
Quinn also mentions the typically limited distribution of who(m) in free relatives that Peter Shor mentioned in his answer:
In Modern English free relatives, the complex wh-forms whoever, whomever, whoso(ever), and whomsoever tend to be favoured over the simplex forms who and whom (cf. Jespersen 1949:62, Baker 1995:210f), except when the relative clause involves VP ellipsis (69) or Null Complement Anaphora (70).