Alpha helices are stabilized because amino acids in the helix can form hydrogen bonds with their amide groups to amide groups directly above and below them in the helix. However, in aqueous solution, these amide bonds would form hydrogen bonds with water, so forming an alpha helix provides little additional stability for the folded state. Therefore, most of the time, if you take a sequence that forms an alpha helix in a protein, isolate it from the rest of the protein, and determine its structure in water, it generally will not form an alpha helix spontaneously. In a lipid environment, however, the unfolded state will not be able to form hydrogen bonds with the solvent, so folding into an alpha helix does provide significant stabilization. This is one reason why the transmembrane regions of membrane proteins are very often alpha helical.
Therefore, the rest of the protein's structure (i.e. the chemical environment around the alpha helix) plays a big role in stabilizing an alpha helix's structure.