Your question happens to sit at the intersection of two subjects I've been fascinated in at an amateur level: the early history of the Mormon Church, Pre-Colombian history, and early American archaeological exploration. The short answer to your question is no, and looking at three decent examples illustrate the issue.

One of the primary, now disproven claims is that the Native Americans were descendants from the tribes of Israel (they were the remnants of a lost tribe, if memory serves). Now, just for a second, ask yourself: how do we know, today, that this claim is untrue? And really, the primary solid evidence we have is genotype testing, that has genetically precluded the possibility that Native Americans are descendants of the Israelites. There are a lot of more superficial arguments you could also use (appearance, language, oral traditions, etc.), but all of those could be overcome by someone slick enough, armed with the purported word of God. It took us cracking genetic testing to really disprove this in a purely scientific way. As an aside, there are other arguments we could use now--e.g., that the Lost Tribe couldn't have accessed the Bering land bridge because it had already closed, but those arguments rely on advancements in knowledge that are remarkably recent--up to the early to mid-20th century. Bill Bryson in A Short History of Nearly Everything has a nice discussion about the surprisingly recently-arriving clarity in terms of the timing of human migration to North America (not a primary source, obv., but he does an excellent job aggregating and summarizing the evolving primary sources from the late 1800s to present).

Arguing in Joseph Smith's favor was the amazing plethora of Native American ruins and works that were scattered across the upper Midwest and New York state (Ohio and NE thereof). Large mounds, substantial village remains, and a reasonably obvious network of passable roads spoke of cultures far more advanced that what Americans of the mid-1800s would have been led to believe by their schools and their interactions with existing tribes/individual Native Americans. And so it wasn't hard to sell a tale of cataclysmic strife--a war in which an intelligent, noble, and accomplished race was eliminated by a base and brutish race. That would explain why the observable archaeological record was so at odds with the "common wisdom" regarding Native Americans at the time. In this way, the people in the geographic sphere where Joseph Smith started (OH, PN, NY) had some personal experiences that would comport with the story Smith was selling.

A more contemporaneous problem is not directly implicated by your question, but provides an interesting insight into how the Mormon Church responded to problems between scientific knowledge and Church doctrine. Joseph Smith had seen (I think at a museum in NYC, but my memory is fuzzy) some Egyptian hieroglyphics depicting what turned out to be a burial preparation scene. Hieroglyphics had not yet been translated, and were (at that time) thought by the general public to be impossible to translate. And so Smith seized on the idea that this particular hieroglyph depicted his Lost Tribe, getting ready to set out on their journey to America. More solid evidence to back up his claim! Smith incorporated the hieroglyphs he had seen into the Mormon Book of Abraham. Unbeknownst to Smith, Thomas Young et al. were already at work using the Rosetta Stone to translate hieroglyphs. When translations become more widely available, and called into serious question the "translation" Smith provided (to put it mildly), the Church offered a series of increasingly dubious defenses. And, apparently, no substantial damage to the Church was done, suggesting that, past a certain point, discrepancies between demonstrable historical fact and Church doctrine are not as problematic as one might think.

Two fairly solid resources to draw on for this subject are 1491, by Charles C. Mann (on the state of the Americas before Columbus) and Fawn Brodie's No Man Knows My History (on Joseph Smith and the early Mormon Church).

Ethan Smith (no relation to Joseph Smith) published View of the Hebrews in 1823 in New England. Joseph Smith wrote the Book of Mormon in 1829 in western New York. This shows me that Europeans were trying to come up with plausible explanations for the existence of the Native Americans that fit into their world view. That world view was based more on the Bible than on science. Ethan Smith's book was first to postulate that the natives might be a remnant tribe of Israel. Joseph Smtih's variant had more scriptural tie ins, including Jesus visting the new World after his crucifixion. The Book of Mormon became the basis around which Smith founded his new religion. This idea that the natives were savages and inferior to the Europeans resonated with that racism. Smith expected the Native Americans to read his book and convert to his church en masse. That didn't happen. He did win converts in industrial/impoverished England and Wales, though.

Modern DNA techniques have only begun to weigh in the question in the last 30 to 40 years. What had been based on myth is now not accepted without proof. Modern humans may have existed for 100k to 200k years, not 6k to 10k per the bible or young earth creationists, or according to mormon theology. Smith's Book of Mormon proposes a timeline that does not correspond to human migration or seafaring ability. Human habitation in the Americas has not been fully examined, but modern estimates go back at least 20k years. A dig at a site in Snowmass, Colorado may push habitation back to 40k. The scientific and religious versions are widely different. This interview with Yale archeology professor Michael D. Coe was very interesting. Coe is an expert on the Mayan civilization and pre-Columbus Meso-America.

Another area where the Book of Mormon is being examined is by digital analysis of Smith's word choices. Since its publication the book has been criticized for its apparent plagiarism of the Bible. A small team of investigators examined books published before the Book of Mormon and looked for similar phrasing. Their analysis pointed strongly to a textbook from the 1820s written by Gilbert Hunt on the War of 1812. A closer analysis of the Late War points to liberal copying of ideas from that book, and others.

The other aspect of the Book of Mormon that is not visited with this question, is the link to the deluge myth and the Tower of Babel.

The tower myth and the resultant distribution of languages is assumed as literal in the Book of Mormon.
This does not comply with even modern linguistic knowledge, and I'd suggest that even as early as the period in which Smith produced the book, scholars were discovering this for themselves.

Remember, the book doesn't just claim that America was populated by seafarers from the middle east once, but twice.

The book also alludes to the notion that the polynesian tribes are also derived from these migrants.