I am Saudi Arabian, being close to to the Arabian Gulf does not mean anything. The act is between Iran and the US, and the act seems to be done.
There is nothing to worry about. Most likely Saudi is safer than wherever you are coming from.
The current conflict is between US and Iran, and is occurring largely on Iraqi territory. Unless it escalates into full-blown war, which both sides appear to be very keen to avoid, it is exceedingly unlikely that this would have any impact on Saudi Arabia.