Ban? The Europeans (Britain in particular) abhorred slavery
You mean the same Britain which up until 1860 happily imported vast quantities of slave-grown cotton from the USA (including proto-Confederate states)?
The British public may have abhorred slavery, but British industrialists were also very good at holding their noses and buying slave-grown cotton. So were the British public prepared to buy textiles made from slave-grown cotton.
and following the Civil War there was a boom in Egyptian and Indian Grown Cotton.
Erm, the short answer to this is "No, utterly wrong."
The longer answer is that it took the sky-high cotton prices to make Egyptian cotton cultivation possible at all. The prices were high enough that they could turn agricultural land over to cotton and import food. Once the ACW ended and cotton prices returned to normal, Egyptian cotton production collapsed. Significant cotton production in Egypt did not return until the 1890s, and it did not become a serious competitor to Southern cotton until the twentieth century.
Indian cotton was of inferior quality to Southern cotton, and had to be sold at a discount. Even then, as Southern cotton exports resumed after the ACW, they continued to gradually displace cotton imports to Britain from India (and elsewhere).
Europe could pretty easily get Cotton from other supporters.
Not in the quality or quantity needed. Egypt's out, India's out, the remnant USA doesn't produce enough supply.
The US still has Tennessee, Missouri, and Northern Arkansas here and could conceivably grow its own (free labor) cotton and I'm sure those producers would lobby hard in favor of tariffs on the CSA's Cotton.
Would this be the same Britain which viewed tariffs as immoral, and which was so committed to free trade that it avoided imposing tariffs even in response to its own colonies putting tariffs on British imports? (As several Australian colonies did, and Britain still refused to put tariffs in retaliation.)
This is not the kind of country which is going to be swayed by American lobbying to impose tariffs on the CSA.