How do I sizeToFit a UILabel by changing only the height and not the width?

You can achieve the same result with sizeThatFits.

CGSize size = [label sizeThatFits:CGSizeMake(label.frame.size.width, CGFLOAT_MAX)];
CGRect frame = label.frame;
frame.size.height = size.height;
label.frame = frame;

Or alternatively, with sizeToFit.

CGRect frame = label.frame;
[label sizeToFit];
frame.size.height = label.frame.size.height;
label.frame = frame;