Those who opposed the custom of women wearing a head covering, as was customary in the middle-east, were those 'inclined to be argumentative'. Paul is not negating any part of his argument but showing how firm his stance was on the issue.
Paul always stressed submitting to local cultures and customs. This verse at the end follows a lengthly argument why in particular rebelling from this middle-eastern custom of the head covering for women was not proper. He even appeals to nature to oppose these radial feminists that were willing to overturn this custom, probably based on the liberty and equality that the gospel gave both slaves and women before God. It would be an abrupt conflict of argument to all of a sudden throw in a statement that overturns his whole preceding argument. Therefore, Paul simply means that he will have no patience for any who support a radical feminist movement that would overturn the customs of the day. Nor should any church recognize that kind of movement.
Even today in many parts of the middle-east this would be rude, unwise and a stumbling block to those who would understand the gesture as a rebellion of woman against the headship of a man within the household. This would be so unlike the Apostle's approach to submission to authorities and customs.
For details of Paul's overall argument refer to a recent related question of the topic here.