65-year-old Mr Tappin from Orpington, was released on $1m bail earlier this week, and has vowed to fight the allegations. He denies conspiring to sell missile batteries to terrorists, believing them to be used in the motor industry, and he says he was part of an FBI sting.
If found guilty, he faces a jail term of up to 35 years.
He told the BBC, "I'm not a terrorist. I've never had any connections with terrorism and I'm just appalled that things could come to this sort of stage - especially in my life now, when I'm 65, been retired for four years and enjoying retirement.
"I didn't know these batteries were for Hawk missiles and too, I didn't know they were destined for Iran."
Mr Tappin was granted bail with several restrictions: he must wear an electronic tag, had to surrender his passport, and will not be allowed to leave El Paso, where his trial is taking place, and Houston, both in Texas.
His emails will also be monitored.
The case of Mr Tappin has pushed the controversy over US-UK extraditions into the limelight. Critics of a treaty drawn up between the two countries say it makes it easier for the US to extradite UK nationals, but much harder the other way around.
However, senior judge Sir Scott Baker looked into the claims, and said the treaty was fair to UK nationals.