Gaps fallacy

Person1: DNA is encoded information, a language with rules like spoken words or a computer code. The cell contains nanomolecular machinery that reads and interprets this information to build the structures of the cell. There is no known material substance, natural law or force of nature that has been observed to give rise to information or a system that can read and interpret it, except the intelligent mind of the human creature. An intelligent mind is therefore a possible explanation for the origin of the DNA information system and a legitimate avenue of scientific enquiry into the origin of life, even if we can't rule out the possibility of discovering new evidence that points to a material cause.

Person 2: It is true that we can't explain the origin of life or DNA with appeal to the theory of evolution because its mechanism is natural selection, which relies on differential reproduction, and the first cell would not have been capable of reproduction. Regardless of this, the lack of evidence in the fossil record, and the fact that no material cause, including natural selection, has ever been observed to give rise to an information system like DNA, we know life arose from non-life because we refuse to to accept the possibility that anything other than matter and the laws and forces of nature exist, and this limits our possible avenues of investigation to explanations based purely on material phenomena.

Do either of these statements commit the god of the gaps fallacy?

Do either of these statements commit the god of the gaps fallacy? 

No. The argument would have to conclude that because we don't know (we lack a natural explanation), then God did it. Positing God as one of the many possible explanations, even the most likely, escapes this fallacy.

I would like to address some of the logical and reasoning errors in the arguments themselves.

1) These are arguments of induction, that ignore the more general induction that no supernatural "explanation" has ever been uncovered for anything—everything we ever explained has been natural. If we are using inductive reasoning, we can't choose which points of induction we use to determine plausibility and which we ignore out of convenience or support of an ideology.

2) " There is no known material substance, natural law or force of nature that has been observed to give rise to information or a system that can read and interpret it, except the intelligent mind of the human creature." - Again, through induction, we can posit that DNA is the result of a "the intelligent mind of the human creature," which of course, is absurd. We don't get to keep the "intelligent mind" part and conveniently dismiss the "human creature" part. Essentially, through induction, we are saying that because every observed instance is X, this instance is likely to be X a well. If we are justified in ditching the "human creature" part of X, we are justified in ditching the "intelligent mind" part as well—at least in terms of our attempt to use induction.

3) "An intelligent mind is therefore a possible explanation for the origin of the DNA..." This is a non sequitur . We can say it is plausible, but nothing argued prior establishes its possibility.

4) "and a legitimate avenue of scientific enquiry(sic) into the origin of life..." No. Until such time as other intelligent minds become testable (through methodological naturalism), the speculation of other intelligent minds as a mechanism is outside the scope of science. Even in a naturalistic sense where aliens seeded earth with life, this would not get us any closer to the origin of life—just the origin of life on earth.

5) "It is true that we can't explain the origin of life or DNA with appeal to the theory of evolution because its mechanism is natural selection..." No. It is true that we can't explain the origin of life with appeal to the theory of evolution because the theory of evolution has absolutely nothing to do with the origin of life. And the theory of gravity cannot explain farts. As for DNA, natural selection does not explain all of evolution, so the premise is not correct (see https://www.nature.com/articles/d41586-019-03061-x as an example).

6) "because we refuse to accept the possibility that anything other than matter and the laws and forces of nature exist..." We should only accept the possibility once it is demonstrated to be possible. The default position is not to claim it is impossible (unless its impossibility has been demonstrated). Regardless, science deals with methodological naturalism. By definition, we can't "investigate" the supernatural because it is outside of all the known methods of investigation. Once something other than matter and the laws and forces of nature are demonstrated to exist, then they will become another tool in the scientist's toolbox. Until such time, it remains a religious or philosophical hypothesis.