Washington did run twice but the first election there were several other people running. He did not run unopposed. What's true is that he was elected unanimously. In the first presidential election there was no popular vote. Electors from each state voted for the two candidates that they most wanted to be president. Washington was picked by 100% of the electors. John Adams received the 2nd most votes (possible because every elector chose two candidates), and thus was made vice president.
If Adams had received the most votes, egotistical as he was, I'm sure he would have taken the job.
If Adams had received the most votes, egotistical as he was, I'm sure he would have taken the job.