South Korean president declared martial law across the country

[ArMaP]

But I thought the entire point of an "executive" officer of a government was to enable the extraordinary act necessary to maintain governance during an emergency.  Of course, democracy implies ratification, but to say that a president cannot declare martial law seems to imply a lack of trust in the executive.

It depends. In Portugal, the president has the power to declare the state of emergency or the state of siege (we don't have martial law) after consulting the government, but the parliament has the power to override the president's decision.