This is a bit disingenuous. The South shipped raw cotton overseas (primarily to England), where it was used to produced finished cotton goods (textiles, clothing, etc.) which were imported to the U.S. under heavy tariffs. The demand for the South’s raw cotton was a “derived demand,” dependent ultimately on the demand for the finished cotton goods produced from it. So, a tariff on imported, finished cotton goods reduced the demand for the South’s raw cotton.
Not necessarily since the U.S. was not the only, or even the largest consumer of the output of the UK textile industry. They also enjoyed the advantage of being the closest and largest supplier of raw cotton. Demand for their produce was high and as a result the U.S. tariff couldn't have had much effect on their exports, if it had any effect at all.
So, why didn't the South respond by building their own textile mills, thus avoiding import tariffs when shipped North to fill the demand for finished goods? Seems they could have undercut the costs of British goods considerably.
Last I looked, you could sew your own shirts, bedsheets, and other products. You didn’t have to import. BTW, if you are truly interested in this, with an open mind, there is a new book by James Huston, “Calculating the Value of the Union.” He shows, using econometric data, that overwhelmingly the tariff was of almost no importance to southerners-—it was all about slavery. Slave capital EXCEEDED that of all the RRs and textile mills in the North put together. With that kind of $$ at stake, it isn’t about a .5% tax on a hunk of yarn.