I'll defer to Dr. Miller on this if he cares to weigh in, or any other professional economist, but my outsider's impression is that in economics as discussed by Romer the situation is more that contradictory theories are being applied to the same domain, without a serious effort to determine experimentally which (if either) is correct.

Sometimes, yes. And at least one school loudly insists that they don't need no stinkin' experimental evidence, because they're actually doing a deductive formal science. In a sign of uncertain health for economics, they are considered heterodox, but not yet laughed out of polite academia.